Statistical dynamics of continuous systems:

perturbative and approximative approaches∗

Dmitri Finkelshtein1 Yuri Kondratiev2 Oleksandr Kutoviy3

February 7, 2014

Abstract

We discuss general concept of Markov statistical dynamics in the con-tinuum. For a class of spatial birth-and-death models, we develop a per-turbative technique for the construction of statistical dynamics. Particu-lar examples of such systems are considered. For the case of Glauber typedynamics in the continuum we describe a Markov chain approximationapproach that gives more detailed information about statistical evolutionin this model.

AMS Subject Classification (2010): 46E30, 82C21, 47D06Keywords: C0-semigroups continuous systems Markov evolution spa-

tial birth-and-death dynamics correlation functions evolution equations

1 Introduction

Dynamics of interacting particle systems appear in several areas of the com-plex systems theory. In particular, we observe a growing activity in the studyof Markov dynamics for continuous systems. The latter fact is motivated, inparticular, by modern problems of mathematical physics, ecology, mathemati-cal biology, and genetics, see e.g. [27, 28, 31–34, 36–39, 51–53, 68] and literaturecited therein. Moreover, Markov dynamics are used for the construction ofsocial, economic and demographic models. Note that Markov processes for con-tinuous systems are considering in the stochastic analysis as dynamical pointprocesses [43, 44, 46] and they appear even in the representation theory of biggroups [10–14].

∗This work was financially supported by the DFG through SFB 701: “Spektrale Strukturenund Topologische Methoden in der Mathematik”

1Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.(d.l.finkelshtein@swansea.ac.uk).

2Fakultat fur Mathematik, Universitat Bielefeld, 33615 Bielefeld, Germany(kondrat@math.uni-bielefeld.de).

3Department of Mathematics, Massachusetts Institute of Technology, 77 MassachusettsAvenue E18-420, Cambridge, MA, USA (kutovyi@mit.edu); Fakultat fur Mathematik, Uni-versitat Bielefeld, 33615 Bielefeld, Germany (kutoviy@math.uni-bielefeld.de).

1

A mathematical formalization of the problem may be described as the fol-lowing. As a phase space of the system we use the space Γ(Rd) of locally finiteconfigurations in the Euclidean space Rd. An heuristic Markov generator whichdescribes considered model is given by its expression on a proper set of func-tions (observables) over Γ(Rd). With this operator we can relate two evolutionequations. Namely, Kolmogorov backward equation for observables and Kol-mogorov forward equation on probability measures on the phase space Γ(Rd)(macroscopic states of the system). The latter equation is a.k.a. Fokker–Planckequation in the mathematical physics terminology. Comparing with the usualsituation in the stochastic analysis, there is an essential technical difficulty: cor-responding Markov process in the configuration space may be constructed onlyin very special particular cases. As a result, a description of Markov dynamics interms of random trajectories is absent for most of models under considerations.

As an alternative approach we use a concept of the statistical dynamics thatsubstitutes the notion of a Markov stochastic process. A central object now isan evolution of states of the system that shall be defined by mean of the Fokker–Planck equation. This evolution equation w.r.t. probability measures on Γ(Rd)may be reformulated as a hierarchical chain of equations for correlation func-tions of considered measures. Such kind of evolution equations are well knownin the study of Hamiltonian dynamics for classical gases as BBGKY chains butnow they appear as a tool for construction and analysis of Markov dynamics.As an essential technical step, we consider related pre-dual evolution chains ofequations on the so-called quasi-observables. As it will be shown in the paper,such hierarchical equations may be analyzed in the framework of semigroup the-ory with the use of powerful techniques of perturbation theory for the semigroupgenerators etc. Considering the dual evolution for the constructed semigroupon quasi-observables we introduce then the dynamics on correlation functions.Described scheme of the dynamics construction looks quite surprising becauseany perturbation techniques for initial Kolmogorov evolution equations one cannot expect. The point is that states of infinite interacting particle systems aregiven by measures which are, in general, mutually orthogonal. As a result, wecan not compare their evolutions or apply a perturbative approach. But underquite general assumptions they have correlation functions and corresponding dy-namics may be considered in a common Banach space of correlation functions.Proper choice of this Banach space means, in fact, that we find an admissibleclass of initial states for which the statistical dynamics may be constructed.There we see again a crucial difference comparing with Markov stochastic pro-cesses framework where the initial distribution evolution is defined for any initialdate.

The structure of the paper is following. In Section 2 we discuss generalconcept of statistical dynamics for Markov evolutions in the continuum andintroduce necessary mathematical structures. Then, in Section 3, this conceptis applied to an important class of Markov dynamics of continuous systems,namely, to birth-and-death models. Here general conditions for the existenceof a semigroup evolution in a space of quasi-observables are obtained. Then weconstruct evolutions of correlation functions as dual objects. It is shown how to

2

apply general results to the study of particular models of statistical dynamicscoming from mathematical physics and ecology.

Finally, in Section 4 we describe an alternative techniques for the construc-tion of solutions to hierarchical chains evolution equations by means of an ap-proximative approach. For concreteness, this approach is discussed in the caseof the so-called Glauber type dynamics in the continuum. We construct a fam-ily of Markov chains on configuration space in finite volumes with concretetransition kernels adopted to the Glauber dynamics. Then the solution to thehierarchical equation for correlation functions may be obtained as the limit ofthe corresponding object for the Markov chain dynamics. This limiting evo-lution generates the state dynamics. Moreover, in the uniqueness regime forthe corresponding equilibrium measure of Glauber dynamics which is, in fact,Gibbs, dynamics of correlation functions is exponentially ergodic.

This paper is based on a series of our previous works [26, 28–30, 34, 53] butcertain results and constructions are detailed and generalized, in particular, inmore complete analysis of the dual dynamics on correlation functions.

2 Statistical description for stochastic dynamicsof complex systems in the continuum

2.1 Complex systems in the continuum

In recent decades, different brunches of natural and life sciences have been ad-dressing to a unifying point of view on a number of phenomena occurring insystems composed of interacting subunits. This leads to formation of an inter-disciplinary science which is referred to as the theory of complex systems. Itprovides reciprocation of concepts and tools involving wide spectrum of appli-cations as well as various mathematical theories such that statistical mechanics,probability, nonlinear dynamics, chaos theory, numerical simulation and manyothers.

Nowadays complex systems theory is a quickly growing interdisciplinary areawith a very broad spectrum of motivations and applications. For instance, hav-ing in mind biological applications, S. Levin [61] characterized complex adaptivesystems by such properties as diversity and individuality of components, local-ized interactions among components, and the outcomes of interactions used forreplication or enhancement of components. We will use a more general in-formal description of a complex system as a specific collection of interactingelements which has so-called collective behavior that means appearance of sys-tem properties which are not peculiar to inner nature of each element itself.The significant physical example of such properties is thermodynamical effectswhich were a background for creation by L. Boltzmann of statistical physics asa mathematical language for studying complex systems of molecules.

We assume that all elements of a complex system are identical by proper-ties and possibilities. Thus, one can model these elements as points in a properspace whereas the complex system will be modeled as a discrete set in this space.

3

Mathematically this means that for the study of complex systems the properlanguage and techniques are delivered by the interacting particle models whichform a rich and powerful direction in modern stochastic and infinite dimensionalanalysis. Interacting particle systems have a wide use as models in condensedmatter physics, chemical kinetics, population biology, ecology (individual basedmodels), sociology and economics (agent based models). For instance a popu-lation in biology or ecology may be represented by a configuration of organismslocated in a proper habitat.

In spite of completely different orders of numbers of elements in real phys-ical, biological, social, and other systems (typical numbers start from 1023 formolecules and, say, 105 for plants) their complexities have analogous phenomenaand need similar mathematical methods. One of them consists in mathemati-cal approximation of a huge but finite real-world system by an infinite systemrealized in an infinite space. This approach was successfully approved to thethermodynamic limit for models of statistical physics and appeared quite usefulfor the ecological modeling in the infinite habitat to avoid boundary effects ina population evolution.

Therefore, our phase space for the mathematical description should consistof countable sets from an underlying space. This space itself may have discreteor continuous nature that leads to segregation of the world of complex systemson two big classes. Discrete models correspond to systems whose elements canoccupy some prescribing countable set of positions, for example, vertices of thelattice Zd or, more generally, of some graph embedded to Rd. These modelsare widely studied and the corresponding theories were realized in numerouspublications, see e.g. [62, 63] and the references therein. Continuous models, ormodels in the continuum, were studied not so intensively and broadly. We con-centrate our attention exactly on continuous models of systems whose elementsmay occupy any points in Eucledian space Rd. (Note that the most part ofour results may be easily transferred to much more general underlying spaces.)Having in mind that real elements have physical sizes we will consider only theso-called locally finite subsets of the underlying space Rd, that means that inany bounded region we assume to have a finite number of the elements. Anotherour restriction will be prohibition of multiple elements in the same position ofthe space.

We will consider systems of elements of the same type only. The mathe-matical realization of considered approaches may be successfully extended tomulti-type systems, meanwhile such systems will have more rich qualitativeproperties and will be an object of interest for applications. Some particularresults can be found e.g. in [21,22,39].

2.2 Mathematical description for a complex systems

We proceed to the mathematical realization of complex systems.Let B(Rd) be the family of all Borel sets in Rd, d ≥ 1; Bb(Rd) denotes the

system of all bounded sets from B(Rd).

4

The configuration space over space Rd consists of all locally finite subsets(configurations) of Rd. Namely,

Γ = Γ(Rd)

:={γ ⊂ Rd

∣∣∣ |γΛ| <∞, for all Λ ∈ Bb(Rd)}.

Here | · | means the cardinality of a set, and γΛ := γ ∩Λ. We may identify eachγ ∈ Γ with the non-negative Radon measure

∑x∈γ δx ∈M(Rd), where δx is the

Dirac measure with unit mass at x,∑x∈∅ δx is, by definition, the zero measure,

and M(Rd) denotes the space of all non-negative Radon measures on B(Rd).This identification allows to endow Γ with the topology induced by the vaguetopology on M(Rd), i.e. the weakest topology on Γ with respect to which allmappings

Γ 3 γ 7→∑x∈γ

f(x) ∈ R (2.1)

are continuous for any f ∈ C0(Rd) that is the set of all continuous functionson Rd with compact supports. It is worth noting the vague topology maybe metrizable in such a way that Γ becomes a Polish space (see e.g. [50] andreferences therein).

Corresponding to the vague topology the Borel σ-algebra B(Γ) appears thesmallest σ-algebra for which all mappings

Γ 3 γ 7→ NΛ(γ) := |γΛ| ∈ N0 := N ∪ {0} (2.2)

are measurable for any Λ ∈ Bb(Rd), see e.g. [1]. This σ-algebra may be generatedby the sets

Q(Λ, n) :={γ ∈ Γ

∣∣ NΛ(γ) = |γΛ| = n}, Λ ∈ Bb(Rd), n ∈ N0. (2.3)

Clearly, for any Λ ∈ Bb(Rd),

Γ =⊔n∈N0

Q(Λ, n).

Among all measurable functions F : Γ→ R := R∪{∞} we mark out the setF0(Γ) consisting of such of them for which |F (γ)| <∞ at least for all |γ| <∞.The important subset of F0(Γ) formed by cylindric functions on Γ. Any such afunction is characterized by a set Λ ∈ Bb(Rd) such that F (γ) = F (γΛ) for allγ ∈ Γ. The class of cylindric functions we denote by Fcyl(Γ) ⊂ F0(Γ).

Functions on Γ are usually called observables. This notion is borrowed fromstatistical physics and means that typically in course of empirical investigationwe may estimate, check, see only some quantities of a whole system rather thenlook on the system itself.

Example 2.1. Let ϕ : Rd → R and consider the so-called linear function on Γ,cf. (2.1),

〈ϕ, γ〉 :=

∑x∈γ

ϕ(x), if∑x∈γ|ϕ(x)| <∞, γ ∈ Γ,

+∞, otherwise.

5

Then, evidently, 〈ϕ, ·〉 ∈ F0(Γ). If, additionally, ϕ ∈ C0(Rd) then 〈ϕ, ·〉 ∈Fcyl(Γ). Not that for e.g. ϕ(x) = ‖x‖Rd (the Euclidean norm in Rd) we havethat 〈ϕ, γ〉 =∞ for any infinite γ ∈ Γ.

Example 2.2. Let φ : Rd\{0} → R be an even function, namely, φ(−x) = φ(x),x ∈ Rd. Then one can consider the so-called energy function

Eφ(γ) :=

∑{x,y}⊂γ

φ(x− y), if∑{x,y}⊂γ

|φ(x− y)| <∞, γ ∈ Γ,

+∞, otherwise.

(2.4)

Clearly, Eφ ∈ F0(Γ). However, even for φ with a compact support, Eφ will notbe a cylindric function.

As we discussed before, any configuration γ represents some system of ele-ments in a real-world application. Typically, investigators are not able to takeinto account exact positions of all elements due to huge number of them. Forquantitative and qualitative analysis of a system researchers mostly need someits statistical characteristics such as density, correlations, spatial structures andso on. This leads to the so-called statistical description of complex systemswhen people study distributions of countable sets in an underlying space in-stead of sets themselves. Moreover, the main idea in Boltzmann’s approachto thermodynamics based on giving up the description in terms of evolutionfor groups of molecules and using statistical interpretation of molecules motionlaws. Therefore, the crucial role for studying of complex systems plays distribu-tions (probability measures) on the space of configurations. In statistical physicsthese measures usually called states that accentuates their role for descriptionof systems under consideration.

We denote the class of all probability measures on(Γ,B(Γ)

)by M1(Γ).

Given a distribution µ ∈ M1(Γ) one can consider a collection of random vari-ables NΛ(·), Λ ∈ Bb(Rd) defined in (2.2). They describe random numbers ofelements inside bounded regions. The natural assumption is that these randomvariables should have finite moments. Thus, we consider the class M1

fm(Γ) ofall measures from M1(Γ) such that∫

Γ

|γΛ|n dµ(γ) <∞, Λ ∈ Bb(Rd), n ∈ N. (2.5)

Example 2.3. Let σ be a non-atomic Radon measure on(Rd,B(Rd)

). Then

the Poisson measure πσ with intensity measure σ is defined on B(Γ) by

πσ(Q(Λ, n)

)=

(σ(Λ)

)nn!

exp{−σ(Λ)

}, Λ ∈ Bb(Rd), n ∈ N0. (2.6)

This formula is nothing but the statement that the random variables NΛ havePoissonian distribution with mean value σ(Λ), Λ ∈ Bb(Rd). Note that by theRenyi theorem [47, 74] a measure πσ will be Poissonian if (2.6) holds for n = 0

6

only. In the case then dσ(x) = ρ(x) dx one can say about nonhomogeneousPoisson measure πρ with density (or intensity) ρ. This notion goes back to thefamous Campbell formula [15,16] which states that∫

Γ

〈ϕ, γ〉 dπρ(γ) =

∫Rdϕ(x)ρ(x) dx, (2.7)

if only the right hand side of (2.7) is well-defined. The generalization of (2.7)is the Mecke identity [65]∫

Γ

∑x∈γ

h(x, γ) dπσ(γ) =

∫Γ

∫Rdh(x, γ ∪ x) dσ(x) dπσ(γ), (2.8)

which holds for all measurable nonnegative functions h : Rd × Γ → R. Hereand in the sequel we will omit brackets for the one-point set {x}. In [65], itwas shown that the Mecke identity is a characterization identity for the Poissonmeasure. In the case ρ(x) = z > 0, x ∈ Rd one can say about the homogeneousPoisson distribution (measure) πz with constant intensity z. We will omit sub-index for the case z = 1, namely, π := π1 = πdx. Note that the property (2.5)is followed from (2.8) easily.

Example 2.4. Let φ be as in Example 2.2 and suppose that the energy given by(2.4) is stable: there exists B ≥ 0 such that, for any |γ| < ∞, Eφ(γ) ≥ −B|γ|.An example of such φ my be given by the expansion

φ(x) = φ+(x) + φp(x), x ∈ Rd, (2.9)

where φ+ ≥ 0 whereas φp is a positive defined function on Rd (the Fouriertransform of a measure on Rd), see e.g. [40, 75]. Fix any z > 0 and definethe Gibbs measure µ ∈ M1(Γ) with potential φ and activity parameter z as ameasure which satisfies the following generalization of the Mecke identity:∫

Γ

∑x∈γ

h(x, γ) dµ(γ) =

∫Γ

∫Rdh(x, γ ∪ x) exp{−Eφ(x, γ)} zdx dµ(γ), (2.10)

where

Eφ(x, γ) := 〈φ(x− ·), γ〉 =∑y∈γ

φ(x− y), γ ∈ Γ, x ∈ Rd \ γ. (2.11)

The identity (2.10) is called the Georgii–Nguyen–Zessin identity, see [45,67]. Ifpotential φ is additionally satisfied the so-called integrability condition

β :=

∫Rd

∣∣e−φ(x) − 1∣∣ dx <∞, (2.12)

then it can checked that the condition (2.5) for the Gibbs measure holds. Notethat under conditions zβ ≤ (2e)−1 there exists a unique measure on

(Γ,B(Γ)

)which satisfies (2.10). Heuristically, the measure µ may be given by the formula

dµ(γ) =1

Ze−E

φ(γ) dπz(γ), (2.13)

7

where Z is a normalizing factor. To give rigorous meaning for (2.13) it is possibleto use the so-called DLR-approach (named after R. L. Dobrushin, O. Lanford,D. Ruelle), see e.g. [2] and references therein. As was shown in [67], this approachgives the equivalent definition of the Gibbs measures which satisfies (2.10).

Note that (2.13) could have a rigorous sense if we restrict our attention onthe space of configuration which belong to a bounded domain Λ ∈ Bb(Rd). Thespace of such (finite) configurations will be denoted by Γ(Λ). The σ-algebraB(Γ(Λ)) may be generated by family of mappings Γ(Λ) 3 γ 7→ NΛ′(γ) ∈ N0,Λ′ ∈ Bb(Rd), Λ′ ⊂ Λ. A measure µ ∈ M1

fm(Γ) is called locally absolutelycontinuous with respect to the Poisson measure π if for any Λ ∈ Bb(Rd) theprojection of µ onto Γ(Λ) is absolutely continuous with respect to (w.r.t.) theprojection of π onto Γ(Λ). More precisely, if we consider the projection mappingpΛ : Γ → Γ(Λ), pΛ(γ) := γΛ then µΛ := µ ◦ p−1

Λ is absolutely continuous w.r.t.πΛ := π ◦ p−1

Λ .

Remark 2.5. Having in mind (2.13), it is possible to derive from (2.10) thatthe Gibbs measure from Example 2.4 is locally absolutely continuous w.r.t. thePoisson measure, see e.g. [24] for the more general case.

By e.g. [48], for any µ ∈M1fm(Γ) which is locally absolutely continuous w.r.t

the Poisson measure there exists the family of (symmetric) correlation functions

k(n)µ : (Rd)n → R+ := [0,∞) which defined as follows. For any symmetric

function f (n) : (Rd)n → R with a finite support the following equality holds∫Γ

∑{x1,...,xn}⊂γ

f (n)(x1, . . . , xn) dµ(γ)

=1

n!

∫(Rd)n

f (n)(x1, . . . , xn)k(n)µ (x1, . . . , xn) dx1 . . . dxn (2.14)

for n ∈ N, and k(0)µ := 1.

The meaning of the notion of correlation functions is the following: the

correlation function k(n)µ (x1, . . . , xn) describes the non-normalized density of

probability to have points of our systems in the positions x1, . . . , xn.

Remark 2.6. Iterating the Mecke identity (2.8), it can be easily shown that

k(n)πρ (x1, . . . , xn) =

n∏i=1

ρ(xi), (2.15)

in particular,k(n)πz (x1, . . . , xn) ≡ zn. (2.16)

Remark 2.7. Note that if potential φ from Example 2.4 satisfies to (2.9), (2.12)then, by [76], there exists C = C(z, φ) > 0 such that for µ defined by (2.10)

k(n)µ (x1, . . . , xn) ≤ Cn, x1, . . . , xn ∈ Rd. (2.17)

The inequality (2.17) is referred to as the Ruelle bound.

8

We dealt with symmetric function of n variables from Rd, hence, they canbe considered as functions on n-point subsets from Rd. We proceed now to theexact constructions.

The space of n-point configurations in Y ∈ B(Rd) is defined by

Γ(n)(Y ) :={η ⊂ Y

∣∣ |η| = n}, n ∈ N.

We put Γ(0)(Y ) := {∅}. As a set, Γ(n)(Y ) may be identified with the sym-metrization of ›Y n =

{(x1, . . . , xn) ∈ Y n

∣∣ xk 6= xl if k 6= l}.

Hence, one can introduce the corresponding Borel σ-algebra, which we denoteby B

(Γ(n)(Y )

). The space of finite configurations in Y ∈ B(Rd) is defined as

Γ0(Y ) :=⊔n∈N0

Γ(n)(Y ). (2.18)

This space is equipped with the topology of the disjoint union. Let B(Γ0(Y )

)denote the corresponding Borel σ-algebra. In the case of Y = Rd we will omitthe index Y in the previously defined notations. Namely,

Γ0 := Γ0(Rd), Γ(n) := Γ(n)(Rd), n ∈ N0. (2.19)

The restriction of the Lebesgue product measure (dx)n to(Γ(n),B(Γ(n))

)we

denote by m(n). We set m(0) := δ{∅}. The Lebesgue–Poisson measure λ on Γ0

is defined by

λ :=∞∑n=0

1

n!m(n). (2.20)

For any Λ ∈ Bb(Rd) the restriction of λ to Γ0(Λ) = Γ(Λ) will be also denotedby λ.

Remark 2.8. The space(Γ,B(Γ)

)is the projective limit of the family of measur-

able spaces{(

Γ(Λ),B(Γ(Λ)))}

Λ∈Bb(Rd). The Poisson measure π on

(Γ,B(Γ)

)from Example 2.3 may be defined as the projective limit of the family ofmeasures {πΛ}Λ∈Bb(Rd), where πΛ := e−m(Λ)λ is the probability measure on(Γ(Λ),B(Γ(Λ))

)and m(Λ) is the Lebesgue measure of Λ ∈ Bb(Rd) (see e.g. [1]

for details).

Functions on Γ0 will be called quasi-observables. Any B(Γ0)-measurablefunction G on Γ0, in fact, is defined by a sequence of functions

{G(n)

}n∈N0

where G(n) is a B(Γ(n))-measurable function on Γ(n). We preserve the samenotation for the function G(n) considered as a symmetric function on (Rd)n.Note that G(0) ∈ R.

A set M ∈ B(Γ0) is called bounded if there exists Λ ∈ Bb(Rd) and N ∈ Nsuch that

M ⊂N⊔n=0

Γ(n)(Λ).

9

The set of bounded measurable functions on Γ0 with bounded support we denoteby Bbs(Γ0), i.e., G ∈ Bbs(Γ0) iff G �Γ0\M= 0 for some bounded M ∈ B(Γ0).

For any G ∈ Bbs(Γ0) the functions G(n) have finite supports in (Rd)n and maybe substituted into (2.14). But, additionally, the sequence of G(n) vanishes forbig n. Therefore, one can summarize equalities (2.14) by n ∈ N0. This leads tothe following definition.

Let G ∈ Bbs(Γ0), then we define the function KG : Γ→ R such that:

(KG)(γ) :=∑ηbγ

G(η) (2.21)

= G(0) +∞∑n=1

∑{x1,...,xn}⊂γ

G(n)(x1, . . . , xn), γ ∈ Γ,

see e.g. [48, 59, 60]. The summation in (2.21) is taken over all finite subcon-figurations η ∈ Γ0 of the (infinite) configuration γ ∈ Γ; we denote this by thesymbol, η b γ. The mapping K is linear, positivity preserving, and invertible,with

(K−1F )(η) :=∑ξ⊂η

(−1)|η\ξ|F (ξ), η ∈ Γ0. (2.22)

By [48], for anyG ∈ Bbs(Γ0), KG ∈ Fcyl(Γ), moreover, there exists C = C(G) >0, Λ = Λ(G) ∈ Bb(Rd), and N = N(G) ∈ N such that

|KG(γ)| ≤ C(1 + |γΛ|

)N, γ ∈ Γ. (2.23)

The expression (2.21) can be extended to the class of all nonnegative mea-surable G : Γ0 → R+, in this case, evidently, KG ∈ F0(Γ). Stress that the lefthand side (l.h.s.) of (2.22) has a meaning for any F ∈ F0(Γ), moreover, in thiscase (KK−1F )(γ) = F (γ) for any γ ∈ Γ0.

For G as above we may summarize (2.14) by n and rewrite the result in acompact form: ∫

Γ

(KG)(γ)dµ(γ) =

∫Γ0

G(η)kµ(η)dλ(η). (2.24)

As was shown in [48], the equality (2.21) may be extended on all functions Gsuch that the l.h.s. of (2.24) is finite. In this case (2.21) holds for µ-a.a. γ ∈ Γand (2.24) holds too.

Remark 2.9. The equality (2.24) may be considered as definition of the corre-lation function kµ. In fact, the definition of correlation functions in statisticalphysics, given by N. N. Bogolyubov in [7], based on a similar relation. More pre-cisely, consider for a B(Rd)-measurable function f the so-called coherent state,given as a function on Γ0 by

eλ(f, η) :=∏x∈η

f(x), η ∈ Γ0\{∅}, eλ(f, ∅) := 1. (2.25)

10

Then for any f ∈ C0(Rd) we have the point-wise equality(Keλ(f)

)(γ) =

∏x∈γ

(1 + f(x)

), η ∈ Γ0. (2.26)

As a result, the correlation functions of different orders may be considered askernels of a Taylor-type expansion∫

Γ

∏x∈γ

(1 + f(x)

)dµ(γ) = 1 +

∞∑n=1

1

n!

∫(Rd)n

n∏i=1

f(xi)k(n)µ (x1, . . . , xn) dx1 . . . dxn

=

∫Γ0

eλ(f, η)kµ(η) dλ(η). (2.27)

Remark 2.10. By (2.18)–(2.20), we have that for any f ∈ L1(Rd, dx)∫Γ0

eλ(f, η)dλ(η) = exp{∫

Rdf(x)dx

}. (2.28)

As a result, taking into account (2.15), we obtain from (2.27) the expression forthe Laplace transform of the Poisson measure∫

Γ

e−〈ϕ,γ〉 dπρ(γ) =

∫Γ0

eλ(e−ϕ(x) − 1, η

)eλ(ρ, η) dλ(η)

= exp{−∫Rd

(1− e−ϕ(x)

)ρ(x)dx

}, ϕ ∈ C0(Rd).

Remark 2.11. Of course, to obtain convergence of the expansion (2.27) for, say,

f ∈ L1(Rd, dx) we need some bounds for the correlation functions k(n)µ . For

example, if the generalized Ruelle bound holds, that is, cf. (2.17),

k(n)µ (x1, . . . , xn) ≤ ACn(n!)1−δ, x1, . . . , xn ∈ Rd (2.29)

for some A,C > 0, δ ∈ (0, 1] independent on n, then the l.h.s. of (2.27) may beestimated by the expression

1 +A∞∑n=1

(C‖f‖L1(Rd)

)n(n!)δ

<∞.

For a given system of functions k(n) on (Rd)n the question about existenceand uniqueness of a probability measure µ on Γ which has correlation functions

k(n)µ = k(n) is an analog of the moment problem in classical analysis. Significant

results in this area were obtained by A. Lenard.

Proposition 2.12 ([58], [60]). Let k : Γ0 → R.1. Suppose that k is a positive definite function, that means that for any

G ∈ Bbs(Γ0) such that (KG)(γ) ≥ 0 for all γ ∈ Γ the following inequality holds∫Γ0

G(η)k(η) dλ(η) ≥ 0. (2.30)

11

Suppose also that k(∅) = 1. Then there exists at least one measure µ ∈M1fm(Γ)

such that k = kµ.2. For any n ∈ N, Λ ∈ Bb(Rd), we set

sΛn :=

1

n!

∫Λnk(n)(x1, . . . , xn) dx1 . . . dxn.

Suppose that for all m ∈ N, Λ ∈ Bb(Rd)∑n∈N

(sΛn+m

)− 1n =∞. (2.31)

Then there exists at most one measure µ ∈M1fm(Γ) such that k = kµ.

Remark 2.13. 1. In [58, 60], the wider space of multiple configurations wasconsidered. The adaptation for the space Γ was realized in [57].

2. It is worth noting also that the growth of correlation functions k(n) up to(n!)2 is admissible to have (2.31).

3. Another conditions for existence and uniqueness for the moment problemon Γ were srudied in [4, 48].

2.3 Statistical descriptions of Markov evolutions

Spatial Markov processes in Rd may be described as stochastic evolutions ofconfigurations γ ⊂ Rd. In course of such evolutions points of configurationsmay disappear (die), move (continuously or with jumps from one position toanother), or new particles may appear in a configuration (that is birth). Therates of these random events may depend on whole configuration that reflect aninteraction between elements of the our system.

The construction of a spatial Markov process in the continuum is highlydifficult question which is not solved in a full generality at present, see e.g.a review [71] and more detail references about birth-and-death processes inSection 3. Meanwhile, for the discrete systems the corresponding processes areconstructed under quite general assumptions, see e.g. [62]. One of the maindifficulties for continuous systems includes the necessity to control number ofelements in a bounded region. Note that the construction of spatial processeson bounded sets from Rd are typically well solved, see e.g. [41].

The existing Markov process Γ 3 γ 7→ Xγt ∈ Γ, t > 0 provides solution the

backward Kolmogorov equation for bounded continuous functions:

∂

∂tFt = LFt, (2.32)

where L is the Markov generator of the process Xt. The question about exis-tence and properties of solutions to (2.32) in proper spaces itself is also highlynontrivial problem of infinite-dimensional analysis. The Markov generator Lshould satisfies the following two (informal) properties: 1) to be conservative,that is L1 = 0, 2) maximum principle, namely, if there exists γ0 ∈ Γ such

12

that F (γ) ≤ F (γ0) for all γ ∈ Γ, then (LF )(γ0) ≤ 0. These properties mightyield that the semigroup, related to (2.32) (provided it exists), will preservesconstants and positive functions, correspondingly.

To consider an example of such L let us consider a general Markov evolutionwith appearing and disappearing of groups of points (giving up the case ofcontinuous moving of particles). Namely, let F ∈ Fcyl(Γ) and set

(LF )(γ) =∑ηbγ

∫Γ0

c(η, ξ, γ \ η)[F ((γ \ η) ∪ ξ)− F (γ)

]dλ(ξ). (2.33)

Heuristically, it means that any finite group η of points from the existing con-figuration γ may disappear and simultaneously a new group ξ of points mayappear somewhere in the space Rd. The rate of this random event is equal toc(η, ξ, γ \ η) ≥ 0. We need some minimal conditions on the rate c to guaranteethat at least

LF ∈ F0(Γ) for all F ∈ Fcyl(Γ) (2.34)

(see Section 3 for a particular case). The term in the sum in (2.33) with η = ∅corresponds to a pure birth of a finite group ξ of points whereas the part ofintegral corresponding to ξ = ∅ (recall that λ({∅}) = 1) is related to pure deathof a finite sub-configuration η ⊂ γ. The parts with |η| = |ξ| 6= 0 corresponds tojumps of one group of points into another positions in Rd. The rest parts presentsplitting and merging effects. In the present paper the technical realization ofthe ideas below is given for one-point birth-and-death parts only, i.e. for thecases |η| = 0, |ξ| = 1 and |η| = 1, |ξ| = 0, correspondingly.

As we noted before, for most cases appearing in applications, the existenceproblem for a corresponding Markov process with a generator L is still open.On the other hand, the evolution of a state in the course of a stochastic dy-namics is an important question in its own right. A mathematical formulationof this question may be realized through the forward Kolmogorov equation forprobability measures (states) on the configuration space Γ. Namely, we considerthe pairing between functions and measures on Γ given by

〈F, µ〉 :=

∫Γ

F (γ) dµ(γ). (2.35)

Then we consider the initial value problem

d

dt〈F, µt〉 = 〈LF, µt〉, t > 0, µt

∣∣t=0

= µ0, (2.36)

where F is an arbitrary function from a proper set, e.g. F ∈ K(Bbs(Γ0)

)⊂

Fcyl(Γ). In fact, the solution to (2.36) describes the time evolution of distribu-tions instead of the evolution of initial points in the Markov process. We rewrite(2.36) in the following heuristic form

∂

∂tµt = L∗µt, (2.37)

13

where L∗ is the (informally) adjoint operator of L with respect to the pairing(2.35).

In the physical literature, (2.37) is referred to the Fokker–Planck equation.The Markovian property of L yields that (2.37) might have a solution in theclass of probability measures. However, the mere existence of the correspondingMarkov process will not give us much information about properties of the solu-tion to (2.37), in particular, about its moments or correlation functions. To dothis, we suppose now that a solution µt ∈M1

fm(Γ) to (2.36) exists and remainslocally absolutely continuous with respect to the Poisson measure π for all t > 0provided µ0 has such a property. Then one can consider the correlation functionkt := kµt , t ≥ 0.

Recall that we suppose (2.34). Then, one can calculate K−1LF using (2.22),and, by (2.24), we may rewrite (2.36) in the following way

d

dt〈〈K−1F, kt〉〉 = 〈〈K−1LF, kt〉〉, t > 0, kt

∣∣t=0

= k0, (2.38)

for all F ∈ K(Bbs(Γ0)

)⊂ Fcyl(Γ). Here the pairing between functions on Γ0 is

given by

〈〈G, k〉〉 :=

∫Γ0

G(η)k(η) dλ(η). (2.39)

Let us recall that then, by (2.20),

〈〈G, k〉〉 =∞∑n=0

1

n!

∫(Rd)n

G(n)(x1, . . . , xn)k(n)(x1, . . . , xn) dx1 . . . dxn,

Next, if we substitute F = KG, G ∈ Bbs(Γ0) in (2.38), we derive

d

dt〈〈G, kt〉〉 = 〈〈LG, kt〉〉, t > 0, kt

∣∣t=0

= k0, (2.40)

for all G ∈ Bbs(Γ0). Here the operator

(LG)(η) := (K−1LKG)(η), η ∈ Γ0

is defined point-wise for all G ∈ Bbs(Γ0) under conditions (2.34). As a result,we are interested in a weak solution to the equation

∂

∂tkt = L∗kt, t > 0, kt

∣∣t=0

= k0, (2.41)

where L∗ is dual operator to L with respect to the duality (2.39), namely,∫Γ0

(LG)(η)k(η) dλ(η) =

∫Γ0

G(η)(L∗k)(η) dλ(η). (2.42)

The procedure of deriving the operator L for a given L is fully combinatorialmeanwhile to obtain the expression for the operator L∗ we need an analog of

14

integration by parts formula. For a difference operator L considered in (2.33)this discrete integration by parts rule is presented in Lemma 3.4 below.

We recall that any function on Γ0 may be identified with an infinite vectorof symmetric functions of the growing number of variables. In this approach,the operator L∗ in (2.41) will be realized as an infinite matrix

(L∗n,m

)n,m∈N0

,

where L∗n,m is a mapping from the space of symmetric functions of n variablesinto the space of symmetric functions of m variables. As a result, instead ofequation (2.36) for infinite-dimensional objects we obtain an infinite system of

equations for functions k(n)t each of them is a function of a finite number of

variables, namely

∂

∂tk

(n)t (x1, . . . , xn) =

(L∗n,mk

(n)t

)(x1, . . . , xn), t > 0, n ∈ N0,

k(n)t (x1, . . . , xn)

∣∣t=0

= k(n)0 (x1, . . . , xn).

(2.43)

Of course, in general, for a fixed n, any equation from (2.43) itself is not closed

and includes functions k(m)t of other orders m 6= n, nevertheless, the system

(2.43) is a closed linear system. The chain evolution equations for k(n)t con-

sists the so-called hierarchy which is an analog of the BBGKY hierarchy forHamiltonian systems, see e.g. [18].

One of the main aims of the present paper is to study the classical solution to(2.41) in a proper functional space. The choice of such a space might be basedon estimates (2.17), or more generally, (2.29). However, even the correlationfunctions (2.16) of the Poisson measures shows that it is rather natural to studythe solutions to the equation (2.41) in weighted L∞-type space of functions withthe Ruelle-type bounds. Integrable correlation functions are not natural for thedynamics on the spaces of locally finite configurations. For example, it is well-known that the Poisson measure πρ with integrable density ρ(x) is concentratedon the space Γ0 of finite configurations (since in this case on can consider Rdinstead of Λ in (2.6)). Therefore, typically, the case of integrable correlationfunctions yields that effectively our stochastic dynamics evolves through finiteconfigurations only. Note that the case of an integrable first order correlationfunction is referred to zero density case in statistical physics.

In the present paper the restrict our attention to the so-called sub-Poissoniancorrelation functions. Namely, for a given C > 0 we consider the followingBanach space

KC :={k : Γ0 → R

∣∣ k · C−|·| ∈ L∞(Γ0, dλ)}

(2.44)

with the norm‖k‖KC := ‖C−|·|k(·)‖L∞(Γ0,λ).

It is clear that k ∈ KC implies, cf. (2.17),∣∣k(η)∣∣ ≤ ‖k‖KC C |η| for λ-a.a. η ∈ Γ0. (2.45)

15

In the following we distinguish two possibilities for a study of the initial valueproblem (2.41). We may try to solve this equation in one space KC . The well-posedness of the initial value problem in this case is equivalent with an existenceof the strongly continuous semigroup (C0-semigroup in the sequel) in the space

KC with a generator L∗. However, the space KC is isometrically isomorphic tothe space L∞(Γ0, C

|·|dλ) whereas, by the H. Lotz theorem [64], [3], in a L∞ spaceany C0-semigroup is uniformly continuous, that is it has a bounded generator.Typically, for the difference operator L given in (2.33), any operator L∗n,m,cf. (2.43), might be bounded as an operator between two spaces of bounded

symmetric functions of n and m variables whereas the whole operator L∗ isunbounded in KC .

To avoid this difficulties we use a trick which goes back to R. Phillips [72].The main idea is to consider the semigroup in L∞ space not itself but as a dualsemigroup T ∗(t) to a C0-semigroup T (t) with a generator A in the pre-dualL1 space. In this case T ∗(t) appears strongly continuous semigroup not on thewhole L∞ but on the closure of the domain of A∗ only.

In our case this leads to the following scheme. We consider the pre-dualBanach space to KC , namely, for C > 0,

LC := L1(Γ0, C

|·|dλ). (2.46)

The norm in LC is given by

‖G‖C :=

∫Γ0

∣∣G(η)∣∣C |η| dλ(η) =

∞∑n=0

Cn

n!

∫(Rd)n

∣∣G(n)(x1, . . . , xn)∣∣ dx1 . . . dxn.

Consider the initial value problem, cf. (2.40), (2.41),

∂

∂tGt = LGt, t > 0, Gt

∣∣t=0

= G0 ∈ LC . (2.47)

Whereas (2.47) is well-posed in LC there exists a C0-semigroup T (t) in LC .Then using Philips’ result we obtain that the restriction of the dual semigroup

T ∗(t) onto Dom(L∗) will be C0-semigroup with generator which is a part of L∗

(the details see in Section 3 below). This provides a solution to (2.41) which

continuously depends on an initial data from Dom(L∗). And after we wouldlike to find a more useful universal subspace of KC which is not depend on theoperator L∗. The realization of this scheme for a birth-and-death operator Lis presented in Section 3 below. As a result, we obtain the classical solutionto (2.41) for t > 0 in a class of sub-Poissonian functions which satisfy theRuelle-type bound (2.45). Of course, after this we need to verify existence anduniqueness of measures whose correlation functions are solutions to (2.41), cf.Proposition 2.12 above. This usually can be done using proper approximationschemes, see e.g. Section 4.

There is another possibility for a study of the initial value problem (2.41)which we will not touch below. Namely, one can consider this evolutional equa-tion in a proper scale of spaces {KC}C∗≤C≤C∗ . In this case we will have typically

16

that the solution is local in time only. More precisely, there exists T > 0 suchthat for any t ∈ [0, T ) there exists a unique solution to (2.41) and kt ∈ KCt forsome Ct ∈ [C∗, C

∗]. We realized this approach in series of papers [5, 25, 37, 38]using the so-called Ovsyannikov method [69, 77, 78]. This method provides lessrestrictions on systems parameters, however, the price for this is a finite timeinterval. And, of course, the question about possibility to recover measures viasolutions to (2.41) should be also solved separately in this case.

3 Birth-and-death evolutions in the continuum

3.1 Microscopic description

One of the most important classes of Markov evolution in the continuum is givenby the birth-and-death Markov processes in the space Γ of all configurationsfrom Rd. These are processes in which an infinite number of individuals exist ateach instant, and the rates at which new individuals appear and some old onesdisappear depend on the instantaneous configuration of existing individuals [46].The corresponding Markov generators have a natural heuristic representation interms of birth and death intensities. The birth intensity b(x, γ) ≥ 0 characterizesthe appearance of a new point at x ∈ Rd in the presence of a given configurationγ ∈ Γ. The death intensity d(x, γ) ≥ 0 characterizes the probability of the eventthat the point x of the configuration γ disappears, depending on the location ofthe remaining points of the configuration, γ \x. Heuristically, the correspondingMarkov generator is described by the following expression, cf. (2.33),

(LF )(γ) :=∑x∈γ

d(x, γ \ x) [F (γ \ x)− F (γ)]

+

∫Rdb(x, γ) [F (γ ∪ x)− F (γ)] dx, (3.1)

for proper functions F : Γ→ R.The study of spatial birth-and-death processes was initiated by C. Preston

[73]. This paper dealt with a solution of the backward Kolmogorov equation(2.32) under the restriction that only a finite number of individuals are alive ateach moment of time. Under certain conditions, corresponding processes existand are temporally ergodic, that is, there exists a unique stationary distribution.Note that a more general setting for birth-and-death processes only requiresthat the number of points in any compact set remains finite at all times. Afurther progress in the study of these processes was achieved by R. Holley andD. Stroock in [46]. They described in detail an analytic framework for birth-and-death dynamics. In particular, they analyzed the case of a birth-and-deathprocess in a bounded region.

Stochastic equations for spatial birth-and-death processes were formulatedin [42], through a spatial version of the time-change approach. Further, in [43],these processes were represented as solutions to a system of stochastic equations,

17

and conditions for the existence and uniqueness of solutions to these equations,as well as for the corresponding martingale problems, were given. Unfortunately,quite restrictive assumptions on the birth and death rates in [43] do not allowan application of these results to several particular models that are interestingfor applications (see e.g. some of examples below).

A growing interest to the study of spatial birth-and-death processes, whichwe have recently observed, is stimulated by (among others) an important rolewhich these processes play in several applications. For example, in spatial plantecology, a general approach to the so-called individual based models was de-veloped in a series of works, see e.g. [8, 9, 17, 66] and the references therein.These models are described as birth-and-death Markov processes in the con-figuration space Γ with specific rates b and d which reflect biological notionssuch as competition, establishment, fecundity etc. Other examples of birth-and-death processes may be found in mathematical physics. In particular, theGlauber-type stochastic dynamics in Γ is properly associated with the grandcanonical Gibbs measures for classical gases. This gives a possibility to studythese Gibbs measures as equilibrium states for specific birth-and-death Markovevolutions [6]. Starting with a Dirichlet form for a given Gibbs measure, one canconsider an equilibrium stochastic dynamics [54]. However, these dynamics givethe time evolution of initial distributions from a quite narrow class. Namely, theclass of admissible initial distributions is essentially reduced to the states whichare absolutely continuous with respect to the invariant measure. Below we con-struct non-equilibrium stochastic dynamics which may have a much wider classof initial states.

This approach was successfully applied to the construction and analysis ofstate evolutions for different versions of the Glauber dynamics [28, 34, 53] andfor some spatial ecology models [26]. Each of the considered models requiredits own specific version of the construction of a semigroup, which takes intoaccount particular properties of corresponding birth and death rates.

In this Section, we realize a general approach considered in Section 2 tothe construction of the state evolution corresponding to the birth-and-deathMarkov generators. We present conditions on the birth and death intensitieswhich are sufficient for the existence of corresponding evolutions as stronglycontinuous semigroups in proper Banach spaces of correlation functions satis-fying the Ruelle-type bounds. Also we consider weaker assumptions on theseintensities which provide the corresponding evolutions for finite time intervalsin scales of Banach spaces as above.

3.2 Expressions for “L and “L∗. Examples of rates b and d

We always suppose that rates d, b : Rd × Γ → [0; +∞] from (3.1) satisfy thefollowing assumptions

d(x, η), b(x, η) > 0, η ∈ Γ0 \ {∅}, x ∈ Rd \ η, (3.2)

d(x, η), b(x, η) <∞, η ∈ Γ0, x ∈ Rd \ η, (3.3)

18

∫M

(d(x, η) + b(x, η)

)dλ(η) <∞, M ∈ B(Γ0) bounded, a.a. x ∈ Rd, (3.4)∫

Λ

(d(x, η) + b(x, η)

)dx <∞, η ∈ Γ0, Λ ∈ Bb(Rd). (3.5)

Proposition 3.1. Let conditions (3.2)–(3.5) hold. The for any G ∈ Bbs(Γ0)and F = KG one has LF ∈ F0(Γ).

Proof. By (2.23), there exist Λ ∈ Bb(Rd), N ∈ N, C > 0 (dependent on G) suchthat ∣∣F (γ \ x)− F (γ)

∣∣ ≤ C11Λ(x)(1 + |γΛ|

)N, x ∈ γ, γ ∈ Γ,∣∣F (γ ∪ x)− F (γ)

∣∣ ≤ C11Λ(x)(2 + |γΛ|

)N, γ ∈ Γ, x ∈ Rd \ γ.

Then, by (3.3), (3.5), for any η ∈ Γ0,

∣∣(LF )(η)∣∣ ≤ C(2 + |ηΛ|

)N(∑x∈ηΛ

d(x, η \ x) +

∫Λ

b(x, η)dx)<∞.

The statement is proved.

We start from the deriving of the expression for L = K−1LK.

Proposition 3.2. For any G ∈ Bbs(Γ0) the following formula holds

(LG)(η) =−∑ξ⊂η

G(ξ)∑x∈ξ

(K−1d(x, · ∪ ξ \ x)

)(η \ ξ)

+∑ξ⊂η

∫Rd

G(ξ ∪ x)(K−1b(x, · ∪ ξ)

)(η \ ξ)dx, η ∈ Γ0.

(3.6)

Proof. First of all, note that, by (3.3) and (2.22), the expressions(K−1b(x, · ∪

ξ))(η) and

(K−1d(x, · ∪ ξ)

)(η) have sense. Recall that G ∈ Bbs(Γ0) implies

F ∈ Fcyl(Γ) ⊂ F0(Γ), then, by (2.21),

F (γ \ x)− F (γ) =∑ηbγ\x

G(η)−∑ηbγ

G(η) =

= −∑ηbγ\x

G(η ∪ x) = −(K(G(· ∪ x)))(γ \ x).(3.7)

In the same way, for x /∈ γ, we derive

F (γ ∪ x)− F (γ) = (K(G(· ∪ x)))(γ). (3.8)

By Proposition 3.1, the values of (LG)(η) are finite, and, by (2.22), one caninterchange order of summations and integration in the following computations,

19

that takes into account (3.7), (3.8):

(LG)(η) =−∑ζ⊂η

(−1)|η\ζ|∑x∈ζ

d(x, ζ \ x)∑ξ⊂ζ\x

G(ξ ∪ x)

+

∫Rd

∑ζ⊂η

(−1)|η\ζ|b(x, ζ)∑ξ⊂ζ

G(ξ ∪ x) dx,

and making substitution ξ′ = ξ ∪ x ⊂ ζ, one may continue

=−∑ζ⊂η

(−1)|η\ζ|∑ξ′⊂ζ

∑x∈ξ′

d(x, ζ \ x)G(ξ′)

+

∫Rd

∑ζ⊂η

(−1)|η\ζ|b(x, ζ)∑ξ⊂ζ

G(ξ ∪ x) dx.

Next, for any measurable H : Γ0 × Γ0 → R, one has∑ζ⊂η

∑ξ⊂ζ

H(ξ, ζ) =∑ξ⊂η

∑ζ⊂ηξ⊂ζ

H(ξ, ζ) =∑ξ⊂η

∑ζ′⊂η\ξ

H(ξ, ζ ′ ∪ ξ).

Using this changing of variables rule, we continue:

(LG)(η) =−∑ξ⊂η

∑ζ′⊂η\ξ

(−1)|η\(ξ∪ζ′)|∑x∈ξ

d(x, ζ ′ ∪ ξ \ x)G(ξ)

+

∫Rd

∑ξ⊂η

∑ζ′⊂η\ξ

(−1)|η\(ζ′∪ξ)|b(x, ζ ′ ∪ ξ)G(ξ ∪ x) dx,

that yields (3.6), using the equality∣∣η \ (ξ ∪ ζ ′)

∣∣ =∣∣(η \ ξ) \ ζ ′∣∣ and (2.22).

Remark 3.3. The initial value problem (2.47) can be considered in the followingmatrix form, cf. (2.43),

∂

∂tG

(n)t (x1, . . . , xn) =

(Ln,mG

(n)t

)(x1, . . . , xn), t > 0, n ∈ N0,

G(n)t (x1, . . . , xn)

∣∣t=0

= G(n)0 (x1, . . . , xn).

The expression (3.6) shows that the matrix above has on the main diagonal

the collection of operators Ln,n, n ∈ N0 which forms the following operator onfunctions on Γ0:

(LdiagG)(η) = −D(η)G(η)+∑y∈η

∫RdG((η\y)∪x

)[b(x, η)−b(x, η\y)

]dx, (3.9)

where the term in the square brackets is equal, by (2.22), to(K−1b(x, · ∪ (η \

y)))({y}). Next, by (3.6), there exist only one non-zero upper diagonal in the

matrix. The corresponding operator is

(LupperG)(η) =

∫RdG(η ∪ x)b(x, η) dx, (3.10)

20

since(K−1b(x, · ∪ η)

)(∅) = b(x, η). The rest part of the expression (3.6) corre-

sponds to the low diagonals.

As we mentioned above, to derive the expression for L∗ we need some discreteanalog of the integration by parts formula. As such, we will use the partial caseof the well-known lemma (see e.g. [56]):

Lemma 3.4. For any measurable function H : Γ0 × Γ0 × Γ0 → R∫Γ0

∑ξ⊂η

H (ξ, η \ ξ, η) dλ (η) =

∫Γ0

∫Γ0

H (ξ, η, η ∪ ξ) dλ (ξ) dλ (η) (3.11)

if at least one side of the equality is finite for |H|.

In particular, if H(ξ, ·, ·) ≡ 0 if only |ξ| 6= 1 we obtain an analog of (2.8),namely,∫

Γ0

∑x∈η

h(x, η \ x, η)dλ(η) =

∫Γ0

∫Rdh(x, η, η ∪ x)dxdλ(η), (3.12)

for any measurable function h : Rd × Γ0 × Γ0 → R such that both sides makesense.

Using this, one can derive the explicit form of L∗.

Proposition 3.5. For any k ∈ Bbs(Γ0) the following formula holds

(L∗k)(η) =−∑x∈η

∫Γ0

k(ζ ∪ η)(K−1d(x, · ∪ η \ x)

)(ζ)dλ(ζ)

+∑x∈η

∫Γ0

k(ζ ∪ (η \ x))(K−1b(x, · ∪ η \ x)

)(ζ)dλ(ζ),

(3.13)

where L∗k is defined by (2.42).

Proof. Using Lemma 3.4, (2.42), (3.6), we obtain for any G ∈ Bbs(Γ0)∫Γ0

G(η)(L∗k

)(η) dλ(η)

=−∫

Γ0

∑ξ⊂η

G(ξ)∑x∈ξ

(K−1d(x, · ∪ ξ \ x)

)(η \ ξ)k(η) dλ(η)

+

∫Γ0

∑ξ⊂η

∫Rd

G(ξ ∪ x)(K−1b(x, · ∪ ξ)

)(η \ ξ) dxk(η) dλ(η)

=−∫

Γ0

∫Γ0

G(ξ)∑x∈ξ

(K−1d(x, · ∪ ξ \ x)

)(η)k(η ∪ ξ) dλ(η) dλ(ξ)

+

∫Γ0

∫Γ0

∫Rd

G(ξ ∪ x)(K−1b(x, · ∪ ξ)

)(η) dxk(η ∪ ξ) dλ(η) dλ(ξ).

21

Applying (3.12) for the second term, we easily obtain the statement. Thecorrectness of using (2.8) and (3.12) follow from the assumptions that G, k ∈Bbs(Γ0), therefore, all integrals over Γ0 will be taken, in fact, over some boundedM ∈ B(Γ0). Then, using (3.4), (3.5), we obtain that the all integrals are fi-nite.

Remark 3.6. Accordingly to Remark 3.3 (or just directly from (3.13)), we havethat the matrix corresponding to (2.43) has the main diagonal given by

(L∗diagk)(η) = −D(η)k(η)

+∑x∈η

∫Rdk((η \ x) ∪ y

)[b(x, (η \ x) ∪ y

)− b(x, η \ x

)]dy, (3.14)

where we have used (3.12). Next, this matrix has only one non zero low diagonal,given by the expression

(L∗lowk)(η) =∑x∈η

k(η \ x)b(x, η \ x). (3.15)

The rest part of expression (3.13) corresponds to the upper diagonals.

Let us consider now several examples of rates b and d which will appear in thefollowing considerations (concrete examples of birth-and-death dynamics, withsuch rates, important for applications will be presented later). As we see from(3.6), (3.13), we always need to calculate expressions like

(K−1a(x, · ∪ ξ)

)(η),

η ∩ ξ = ∅, where a equal to b or d. We consider the following kinds of functiona : Rd × Γ→ R:

• Constant rate:a(x, γ) ≡ m > 0. (3.16)

If we substitute f ≡ 0 into (2.26), we obtain that

(K−1m)(η) = m0|η|, η ∈ Γ0, (3.17)

where as usual 00 := 1, and, of course, in this case K−1a(x, · ∪ ξ)(η) alsoequal to m0|η| for any ξ ∈ Γ0;

• Linear rate:a(x, γ) = 〈c(x− ·), γ〉 =

∑y∈γ

c(x− y), (3.18)

where c is a potential like in Example 2.2. Any such c for a given x ∈ Rddefines a function Cx : Γ0 → R such that Cx(η) = 0 for all η /∈ Γ(1) and,for any η ∈ Γ(1), y ∈ Rd with η = {y}, we have Cx(η) = c(x − y). Then,in this case, taking into account (3.17) and the obvious equality

〈c(x− ·), η ∪ ξ〉 = 〈c(x− ·), η〉+ 〈c(x− ·), ξ〉, (3.19)

we obtain(K−1a(x, · ∪ ξ)

)(η) = a(x, ξ)0|η| + Cx(η), η ∈ Γ0. (3.20)

22

• Exponential rate:

a(x, γ) = e〈c(x−·),γ〉 = exp{∑y∈γ

c(x− y)}, (3.21)

where c as above. Taking into account (3.19) and (2.26), we obtain thatin this case(

K−1a(x, · ∪ ξ))(η) = a(x, ξ)eλ

(ec(x−·) − 1, η

), η ∈ Γ0. (3.22)

• Product of linear and exponential rates:

a(x, γ) = 〈c1(x− ·), γ〉e〈c2(x−·),γ〉, (3.23)

where c1 and c2 are potentials as before. Then we have

a(x, η ∪ ξ) = a(x, η)e〈c2(x−·),ξ〉 + a(x, ξ)e〈c2(x−·),η〉. (3.24)

Next, by (2.22),(K−1a(x, ·)

)(η) =

∑ζ⊂η

(−1)|η\ζ|∑y∈ζ

c1(x− y)e〈c2(x−·),ζ〉

=∑y∈η

c1(x− y)∑ζ⊂η\y

(−1)|(η\y)\ζ|e〈c2(x−·),ζ∪y〉,

and taking into account (2.26),

=∑y∈η

c1(x− y)ec2(x−y)eλ(ec2(x−·) − 1, η \ y

). (3.25)

By (3.24) and (3.25), we finally obtain that in this case(K−1a(x, · ∪ ξ)

)(η) = e〈c2(x−·),ζ〉

∑y∈η

c1(x− y)ec2(x−y)eλ(ec2(x−·) − 1, η \ y

)+ a(x, ξ)eλ

(ec2(x−·) − 1, η

), η ∈ Γ0.

(3.26)

• Mixing of linear and exponential rates:

a(x, γ) =∑y∈γ

c1(x− y)e〈c2(y−·),γ\y〉. (3.27)

We have

a(x, η ∪ ξ) =∑y∈η

c1(x− y)e〈c2(y−·),η\y〉e〈c2(y−·),ξ〉

+∑y∈ξ

c1(x− y)e〈c2(y−·),η〉e〈c2(y−·),ξ\y〉.

23

Then, similarly to (3.26), we easily derive(K−1a(x, · ∪ ξ)

)(η) =

∑y∈η

c1(x− y)eλ(ec2(y−·) − 1, η \ y

)e〈c2(y−·),ξ〉

+∑y∈ξ

c1(x− y)eλ(ec2(y−·) − 1, η

)e〈c2(y−·),ξ\y〉.

Using the similar arguments one can consider polynomials rates and theircompositions with exponents as well.

3.3 Semigroup evolutions in the space of quasi-observables

We proceed now to the construction of a semigroup in the space LC , C > 0,see (2.46), which has a generator, given by L, with a proper domain. To definesuch domain, let us set

D (η) :=∑x∈η

d (x, η \ x) ≥ 0, η ∈ Γ0; (3.28)

D := {G ∈ LC | D (·)G ∈ LC} . (3.29)

Note that Bbs(Γ0) ⊂ D and Bbs(Γ0) is a dense set in LC . Therefore, D is also a

dense set in LC . We will show now that (L,D) given by (3.6), (3.29) generatesC0-semigroup on LC if only ‘the full energy of death’, given by (3.28), is bigenough.

Theorem 3.7. Suppose that there exists a1 ≥ 1, a2 > 0 such that for all ξ ∈ Γ0

and a.a. x ∈ Rd∑x∈ξ

∫Γ0

∣∣K−1d (x, · ∪ ξ \ x)∣∣ (η)C |η|dλ (η) ≤ a1D(ξ), (3.30)

∑x∈ξ

∫Γ0

∣∣K−1b (x, · ∪ ξ \ x)∣∣ (η)C |η|dλ (η) ≤ a2D(ξ). (3.31)

and, moreover,

a1 +a2

C<

3

2. (3.32)

Then (L,D) is the generator of a holomorphic semigroup T (t) on LC .

Remark 3.8. Having in mind Remark 3.3 one can say that the idea of the proof isto show that the multiplication part of the diagonal operator (3.9) will dominates

on the rest part of the operator matrix(Ln,m

)provided the conditions (3.30),

(3.31) hold. Note also that, by (2.20), (2.18), (2.19), (3.28), the l.h.s of (3.30)is equal to

D(ξ) +∑x∈ξ

∫Γ0\{∅}

∣∣K−1d (x, · ∪ ξ \ x)∣∣ (η)C |η|dλ (η) .

This is the reason to demand that a1 should be not less than 1.

24

Proof of Theorem 3.7. Let us consider the multiplication operator (L0,D) onLC given by

(L0G)(η) = −D (η)G(η), G ∈ D, η ∈ Γ0. (3.33)

We recall that a densely defined closed operators A on LC is called sectorial ofangle ω ∈ (0, π2 ) if its resolvent set ρ(A) contains the sector

Sect(π

2+ ω

):={z ∈ C

∣∣∣ | arg z| < π

2+ ω

}\ {0}

and for each ε ∈ (0;ω) there exists Mε ≥ 1 such that

||R(z,A)|| ≤ Mε

|z|(3.34)

for all z 6= 0 with | arg z| ≤ π

2+ ω − ε. Here and below we will use notation

R(z,A) := (z11−A)−1, z ∈ ρ(A).

The set of all sectorial operators of angle ω ∈ (0, π2 ) in LC we denote by HC(ω).Any A ∈ HC(ω) is a generator of a bounded semigroup T (t) which is holomor-phic in the sector | arg t| < ω (see e.g. [19, Theorem II.4.6]). One can prove thefollowing lemma.

Lemma 3.9. The operator (L0,D) given by (3.33) is a generator of a contrac-tion semigroup on LC . Moreover, L0 ∈ HC(ω) for all ω ∈ (0, π2 ) and (3.34)holds with Mε = 1

cosω for all ε ∈ (0;ω).

Proof of Lemma 3.9. It is not difficult to show that the densely defined operatorL0 is closed in LC . Let 0 < ω < π

2 be arbitrary and fixed. Clear, that for allz ∈ Sect

(π2 + ω

) ∣∣D (η) + z∣∣ > 0, η ∈ Γ0.

Therefore, for any z ∈ Sect(π2 + ω

)the inverse operator R(z, L0) = (z11−L0)−1,

the action of which is given by(R(z, L0)G

)(η) =

1

D (η) + zG(η), (3.35)

is well defined on the whole space LC . Moreover,

|D(η) + z| =»

(D(η) + Re z)2 + (Im z)2 ≥®|z|, if Re z ≥ 0

|Im z|, if Re z < 0,

and for any z ∈ Sect(π2 + ω

)|Im z| = |z|| sin arg z| ≥ |z|

∣∣∣sin(π2

+ ω)∣∣∣ = |z| cosω.

25

As a result, for any z ∈ Sect(π2 + ω

)||R(z, L0)|| ≤ 1

|z| cosω, (3.36)

that implies the second assertion. Note also that |D(η)+z| ≥ Re z for Re z > 0,hence,

||R(z, L0)|| ≤ 1

Re z, (3.37)

that proves the first statement by the classical Hille–Yosida theorem.

For any G ∈ Bbs(Γ0) we define

(L1G) (η) := (LG)(η)− (L0G)(η)

=−∑ξ(η

G(ξ)∑x∈ξ

(K−1d(x, · ∪ ξ \ x)

)(η \ ξ)

+∑ξ⊂η

∫Rd

G(ξ ∪ x)(K−1b(x, · ∪ ξ)

)(η \ ξ)dx.

(3.38)

Next Lemma shows that, under conditions (3.30), (3.31) above, the operatorL1 is relatively bounded by the operator L0.

Lemma 3.10. Let (3.30), (3.31) hold. Then (L1,D) is a well-defined operatorin LC such that

‖L1R(z, L0)‖ ≤ a1 − 1 +a2

C, Re z > 0 (3.39)

and‖L1G‖ ≤

(a1 − 1 +

a2

C

)‖L0G‖, G ∈ D. (3.40)

Proof of Lemma 3.10. By Lemma 3.4, we have for any G ∈ LC , Re z > 0∫Γ0

∣∣∣∣−∑ξ(η

1

z +D(ξ)G(ξ)

∑x∈ξ

(K−1d(x, · ∪ ξ \ x)

)(η \ ξ)

∣∣∣∣C |η|dλ (η)

≤∫

Γ0

∑ξ(η

1

|z +D(ξ)||G(ξ)|

∑x∈ξ

∣∣K−1d(x, · ∪ ξ \ x)∣∣(η \ ξ)C |η|dλ (η)

=

∫Γ0

1

|z +D(ξ)||G(ξ)|

∑x∈ξ

∫Γ0

∣∣K−1d(x, · ∪ ξ \ x)∣∣(η)C |η|dλ (η)C |ξ|dλ (ξ)

−∫

Γ0

1

|z +D(η)|D (η) |G(η)|C |η|dλ (η)

≤(a1 − 1)

∫Γ0

1

Re z +D(η)D(η)|G(η)|C |η|dλ(η) ≤ (a1 − 1)‖G‖C ,

26

and∫Γ0

∣∣∣∣∑ξ⊂η

∫Rd

1

z +D(ξ ∪ x)G(ξ ∪ x)

(K−1b(x, · ∪ ξ)

)(η \ ξ)dx

∣∣∣∣C |η|dλ (η)

≤∫

Γ0

∫Γ0

∫Rd

1

|z +D(ξ ∪ x)||G(ξ ∪ x)|

∣∣K−1b(x, · ∪ ξ)∣∣ (η)dxC |η|C |ξ|dλ (ξ) dλ (η)

=1

C

∫Γ0

1

|z +D(ξ)||G(ξ)|

∑x∈ξ

∫Γ0

∣∣K−1b(x, · ∪ ξ \ x)∣∣ (η)C |η|dλ (η)C |ξ|dλ (ξ)

≤a2

C

∫Γ0

1

Re z +D(ξ)|G(ξ)|D(ξ)C |ξ|dλ (ξ) ≤ a2

C‖G‖C .

Combining these inequalities we obtain (3.39). The same considerations yield∫Γ0

∣∣∣∣−∑ξ(η

G(ξ)∑x∈ξ

(K−1d(x, · ∪ ξ \ x)

)(η \ ξ)

∣∣∣∣C |η|dλ (η)

+

∫Γ0

∣∣∣∣∑ξ⊂η

∫Rd

G(ξ ∪ x)(K−1b(x, · ∪ ξ)

)(η \ ξ)dx

∣∣∣∣C |η|dλ (η)

≤(

(a1 − 1) +a2

C

)∫Γ0

|G(η)|D(η)C |η|dλ (η) ,

that proves (3.40) as well.

And now we proceed to finish the proof of the Theorem 3.7. Let us set

θ := a1 +a2

C− 1 ∈

(0;

1

2

).

Then θ1−θ ∈ (0; 1). Let ω ∈

(0; π2

)be such that cosω < θ

1−θ . Then, by the

proof of Lemma 3.9, L0 ∈ HC(ω) and ||R(z, L0)|| ≤ M|z| for all z 6= 0 with

| arg z| ≤ π

2+ ω, where M := 1

cosω . Then

θ =1

1 + 1−θθ

<1

1 + 1cosω

=1

1 +M.

Hence, by (3.40) and the proof of [19, Theorem III.2.10], we have that (L =L0 + L1,D) is a generator of holomorphic semigroup on LC .

Remark 3.11. By (3.28), the estimates (3.30), (3.31) are satisfied if∫Γ0

∣∣K−1d (x, · ∪ ξ)∣∣ (η)C |η|dλ (η) ≤a1d (x, ξ) , (3.41)∫

Γ0

∣∣K−1b (x, · ∪ ξ)∣∣ (η)C |η|dλ (η) ≤a2d (x, ξ) . (3.42)

27

3.4 Evolutions in the space of correlation functions

In this Subsection we will use the semigroup T (t) acting oh the space of quasi-observables for a construction of solution to the evolution equation (2.41) onthe space of correlation functions.

We denote dλC := C |·|dλ; and the dual space (LC)′ =(L1(Γ0, dλC)

)′=

L∞(Γ0, dλC). As was mentioned before the space (LC)′ is isometrically isomor-phic to the Banach space KC considered in (2.44)–(2.45). The isomorphism isgiven by the isometry RC

(LC)′ 3 k 7−→ RCk := k · C |·| ∈ KC . (3.43)

Recall, one may consider the duality between the Banach spaces LC and KCgiven by (2.39) with

|〈〈G, k〉〉| ≤ ‖G‖C · ‖k‖KC .

Let(L′,Dom(L′)

)be an operator in (LC)′ which is dual to the closed oper-

ator(L,D

). We consider also its image on KC under the isometry RC . Namely,

let L∗ = RCL′RC−1 with the domain Dom(L∗) = RCDom(L′). Similarly, one

can consider the adjoint semigroup T ′(t) in (LC)′ and its image T ∗(t) in KC .

The space LC is not reflexive, hence, T ∗(t) is not C0-semigroup in wholeKC . By e.g. [19, Subsection II.2.5], the last semigroup will be weak*-continuous,

weak*-differentiable at 0 and L∗ will be weak*-generator of T ∗(t). Therefore,one has an evolution in the space of correlation functions. In fact, we havea solution to the evolution equation (2.41), in a weak*-sense. This subsectionis devoted to the study of a classical solution to this equation. By e.g. [19,

Subsection II.2.6], the restriction T�(t) of the semigroup T ∗(t) onto its invariant

Banach subspace Dom(L∗) (here and below all closures are in the norm of the

space KC) is a strongly continuous semigroup. Moreover, its generator L� will

be a part of L∗, namely,

Dom(L�) ={k ∈ Dom(L∗)

∣∣∣ L∗k ∈ Dom(L∗)}

(3.44)

and L�k = L∗k for any k ∈ Dom(L�).

Proposition 3.12. Let (3.30), (3.31) be satisfied. Suppose that there existsA > 0, N ∈ N0, ν ≥ 1 such that for ξ ∈ Γ0 and x /∈ ξ

d (x, ξ) ≤ A(1 + |ξ|)Nν|ξ|, (3.45)

Then for any α ∈(0; 1

ν

)KαC ⊂ Dom(L∗). (3.46)

Proof. In order to show (3.46) it is enough to verify that for any k ∈ KαC there

exists k∗ ∈ KC such that for any G ∈ Dom(L)⟨⟨LG, k

⟩⟩= 〈〈G, k∗〉〉 . (3.47)

28

By the same calculations as in the proof of Proposition 3.5, it is easy to see that(3.47) is valid for any k ∈ KαC with k∗ = L∗k, where L∗ is given by (3.13),provided k∗ ∈ KC .

To prove the last inclusion, one can estimate, by (3.30), (3.31), (3.45), that

C−|η|∣∣∣(L∗k)(η)

∣∣∣≤C−|η|

∑x∈η

∫Γ0

|k(ζ ∪ η)|∣∣K−1d(x, · ∪ η \ x)

∣∣(ζ)dλ(ζ)

+ C−|η|∑x∈η

∫Γ0

|k(ζ ∪ (η \ x))|∣∣K−1b(x, · ∪ η \ x)

∣∣(ζ)dλ(ζ)

≤‖k‖KαC α|η|∑x∈η

∫Γ0

(αC)|ζ| ∣∣K−1d(x, · ∪ η \ x)

∣∣(ζ)dλ(ζ)

+1

αC‖k‖KαC α

|η|∑x∈η

∫Γ0

(αC)|ζ| ∣∣K−1b(x, · ∪ η \ x)

∣∣(ζ)dλ(ζ)

≤ ‖k‖KαC(a1 +

a2

αC

)α|η|

∑x∈η

d (x, η \ x)

≤A ‖k‖KαC(a1 +

a2

αC

)α|η|(1 + |η|)N+1ν|η|−1.

Using elementary inequality

(1 + t)bat ≤ 1

a

Åb

−e ln a

ãb, b ≥ 1, a ∈ (0; 1) , t ≥ 0, (3.48)

we have for αν < 1

ess supη∈Γ0

C−|η|∣∣∣(L∗k)(η)

∣∣∣ ≤ ‖k‖KαC (a1 +a2

αC

) A

αν2

ÅN + 1

−e ln (αν)

ãN+1

<∞.

The statement is proved.

Lemma 3.13. Let (3.45) holds. We define for any α ∈ (0; 1)

Dα : = {G ∈ LαC | D (·)G ∈ LαC} .

Then for any α ∈ (0; 1ν )

D ⊂ LC ⊂ Dα ⊂ LαC (3.49)

Proof. The first and last inclusions are obvious. To prove the second one, weuse (3.45), (3.48) and obtain for any G ∈ LC∫

Γ0

D (η) |G (η)| (αC)|η|dλ (η) ≤

∫Γ0

α|η|∑x∈η

A(1 + |η|)Nν|η|−1 |G (η)|C |η|dλ (η)

≤ const

∫Γ0

|G (η)|C |η|dλ (η) <∞.

The statement is proved.

29

Proposition 3.14. Let (3.30), (3.31), and (3.45) hold with

a1 +a2

αC<

3

2(3.50)

for some α ∈ (0; 1). Then (L,Dα) is a generator of a holomorphic semigroup

Tα (t) on LαC .

Proof. The proof is similar to the proof of Theorem 3.7, taking into accountthat bounds (3.31), (3.30) imply the same bounds for αC instead of C. Notealso that (3.50) is stronger than (3.32).

Proposition 3.15. Let (3.30), (3.31), and (3.45) hold with

1 ≤ ν < C

a2

Å3

2− a1

ã. (3.51)

Then, for any α witha2

C(

32 − a1

) < α <1

ν, (3.52)

the set KαC is a T�(t)-invariant Banach subspace of KC . Moreover, the set

KαC is T�(t)-invariant too.

Proof. First of all note that the condition on α implies (3.50). Next, we prove

that Tα (t)G = T (t)G for any G ∈ LC ⊂ LαC . Let Lα = (L,Dα) is the operator

in LαC . There exists ω > 0 such that (ω; +∞) ⊂ ρ(L) ∩ ρ(Lα), see e.g. [19,

Section III.2]. For some fixed z ∈ (ω; +∞) we denote by R(z, L) =(z11− L

)−1

the resolvent of (L,D) in LC and by R(z, Lα) =(z11 − Lα

)−1the resolvent of

Lα in LαC . Then for any G ∈ LC we have R(z, L)G ∈ D ⊂ Dα and

R(z, L)G−R(z, Lα)G = R(z, Lα)((z11− Lα)−

(z11− L)

)R(z, L)G = 0,

since Lα = L on D. As a result, Tα (t)G = T (t)G on LC .Note that for any G ∈ LC ⊂ LαC and for any k ∈ KαC ⊂ KC we have

Tα(t)G ∈ LαC and ¨Tα(t)G, k

∂∂=¨G, T ∗α(t)k

∂∂,

where, by the same construction as before, T ∗α(t)k ∈ KαC . But G ∈ LC , k ∈ KCimplies ¨

Tα(t)G, k∂∂

=¨T (t)G, k

∂∂=¨G, T ∗(t)k

∂∂.

Hence, T ∗(t)k = T ∗α(t)k ∈ KαC that proves the statement due to continuity of

the family T ∗(t).

30

By e.g. [19, Subsection II.2.3], one can consider the restriction T�α(t) of the

semigroup T�(t) onto KαC . It will be strongly continuous semigroup with the

generator L�α which is a restriction of L� onto KαC . Namely, cf. 3.44,

Dom(L�α) ={k ∈ KαC

∣∣∣ L∗k ∈ KαC}, (3.53)

and L�αk = L�k = L∗k for any k ∈ KαC . In the other words, L�α is a part ofL∗.

And now we may proceed to the main statement of this Subsection.

Theorem 3.16. Let (3.30), (3.31), (3.45), and (3.51) hold, and let α be chosenas in (3.52). Then for any k0 ∈ KαC there exists a unique classical solution

to (2.41) in the space KαC , and this solution is given by kt = T�α(t)k0. More-over, k0 ∈ KαC implies kt ∈ KαC .

Proof. We recall that (L,D) is a densely defined closed operator on LC (as a

generator of a C0-semigroup T (t)). Then, by e.g. [79, Lemma 1.4.1], for the

dual operator(L∗,Dom(L∗

)we have that ρ(L∗) = ρ(L) and, for any z ∈ ρ(L),

R(z, L∗) = R(z, L)∗. In particular,∥∥R(z, L∗)∥∥ =

∥∥R(z, L)∗∥∥ =

∥∥R(z, L)∥∥. (3.54)

Next, if we denote by R(z, L)� the restriction of R(z, L)∗ onto R(z, L)∗-invariant

space Dom(L∗)

then, by e.g. [79, Theorem 1.4.2], ρ(L�) = ρ(L∗) and, for any

z ∈ ρ(L∗) = ρ(L), R(z, L�) = R(z, L)�. Therefore, by (3.54),∥∥R(z, L�)∥∥ ≤ ∥∥R(z, L)

∥∥.Then, taking into account that by Theorem 3.7 the operator (L,D) is a generator

of the holomorphic semigroup T (t), we immediately conclude that the same

property has the semigroup T�(t) with the generator(L�,Dom(L�)

)in the

space Dom(L∗).

As a result, by e.g. [70, Corollary 4.1.5], the initial value problem (2.41) in the

Banach space Dom(L∗)

has a unique classical solution for any k0 ∈ Dom(L∗). In

particular, it means that the solution kt = T�(t)k0 is continuously differentiable

in t w.r.t. the norm of Dom(L∗)

that is the norm ‖·‖KC , and also kt ∈ Dom(L�).

But by Proposition 3.15, the space KαC is T�(t)-invariant. Hence, if we consider

now the initial value k0 ∈ KαC ⊂ Dom(L∗)

we obtain with a necessity that kt =

T�(t)k0 = T�α(t)k0 ∈ KαC . Therefore, kt ∈ KαC⋂

Dom(L�) = Dom(L�α)(see again [19, Subsection II.2.3]) and, recall, kt is continuously differentiable in tw.r.t. the norm ‖ ·‖KC that is the norm in KαC . This completes the proof of thefirst statement. The second one follows directly now from Proposition 3.15.

31

3.5 Examples of dynamics

We proceed now to describing the concrete birth-and-death dynamics whichare important for different application. We will consider the explicit conditionson parameters of systems which imply the general conditions on rates b and dabove. For simplicity of notations we denote the l.h.s. of (3.30) and (3.31) byId(ξ) and Ib(ξ), ξ ∈ Γ0, correspondingly.

Example 3.17. (Surgailis dynamics) Let the rates d and b are independent onconfiguration variable, namely,

d(x, γ) = m(x), b(x, γ) = z(x), x ∈ Rd, γ ∈ Γ, (3.55)

where 0 < m, z ∈ L∞(Rd). Then, by (3.17) we obtain that

Id(ξ) = 〈m, ξ〉 = D(ξ), Ib(ξ) = 〈z, ξ〉, ξ ∈ Γ0.

Therefore, (3.30), (3.31), (3.32) hold if only

z(x) ≤ am(x), x ∈ Rd (3.56)

with any

0 < a <C

2. (3.57)

Clearly, in this case (3.45) holds with N = 0, ν = 1, therefore, the condition(3.52) is just

2a

C< α < 1. (3.58)

The case of constant (in space) m and σ was considered in [23]. Similarly tothat results, one can derive the explicit expression for the solution to the initialvalue problem (2.41) considered point-wise in Γ0, namely,

kt(η) = eλ(e−tm, η)∑ξ⊂η

eλ

( zm

(etm − 1

), ξ)k0(η \ ξ), η ∈ Γ0. (3.59)

Note that, using (3.59), it can be possible to show directly that the statementof Theorem 3.16 still holds if we drop 2 in (3.57) and (3.58).

Example 3.18. (Glauber-type dynamics). Let L be given by (3.1) with

d(x, γ \ x) = m(x) exp{s∑y∈γ\x

φ(x− y)}, x ∈ γ, γ ∈ Γ, (3.60)

b(x, γ) = z(x) exp{

(s− 1)∑y∈γ

φ(x− y)}, x ∈ Rd \ γ, γ ∈ Γ, (3.61)

where φ : Rd \{0} → R+ is a pair potential, φ(−x) = φ(x), 0 < z,m ∈ L∞(Rd),and s ∈ [0; 1]. Note that in the case m(x) ≡ 1, z(x) ≡ z > 0 and for anys ∈ [0; 1] the operator L is well defined and, moreover, symmetric in the space

32

L2(Γ, µ), where µ is a Gibbs measure, given by the pair potential φ and activityparameter z (see e.g. [55] and references therein). This gives possibility tostudy the corresponding semigroup in L2(Γ, µ). If, additionally, s = 0, thecorresponding dynamics was also studied in another Banach spaces, see e.g.[28, 34, 53]. Below we show that one of the main result of the paper stated inTheorem 3.16 can be applied to the case of arbitrary s ∈ [0; 1] and non-constantm and z. Set

βτ :=

∫Rd

∣∣eτφ(x) − 1∣∣dx ∈ [0;∞], τ ∈ [−1; 1]. (3.62)

Let s be arbitrary and fixed. Suppose that βs <∞, βs−1 <∞. Then, by (3.60),(3.61), (3.22), and (2.28),

Id(ξ) =∑x∈ξ

d(x, ξ \ x)eCβs = D(ξ)eCβs ,

and, analogously, taking into account that φ ≥ 0,

Ib(ξ) =∑x∈ξ

b(x, ξ \ x)eCβs−1 ≤∑x∈ξ

z(x)

m(x)d(x, ξ \ x)eCβs−1

Therefore, to apply Theorem 3.7 we should assume that there exists σ > 0 suchthat

z(x) ≤ σm(x), x ∈ Rd, (3.63)

and

eCβs +σ

CeCβs−1 <

3

2. (3.64)

In particular, for s = 0 we need

σ

CeCβ−1 <

1

2. (3.65)

Next, to have (3.45) and (3.51), we will distinguish two cases. For s = 0 weobtain (3.45) since m ∈ L∞(Rd). In this case, ν = 1 that fulfilles (3.51) as well.For s ∈ (0, 1], we should assume that

0 ≤ φ ∈ L∞(Rd). (3.66)

Then, by (3.60), ν = esφ ≥ 1, where φ := ‖φ‖L∞(Rd). Therefore, to have (3.51),we need the following improvement of (3.64):

eCβs +σ

Cesφ+Cβs−1 <

3

2. (3.67)

Example 3.19. (Bolker–Dieckman–Law–Pacala (BDLP) model) This exampledescribes a generalization of the model of plant ecology (see [26] and references

33

therein). Let L be given by (3.1) with

d(x, γ \ x) = m(x) + κ−(x)∑y∈γ\x

a−(x− y), x ∈ γ, γ ∈ Γ, (3.68)

b(x, γ) = κ+(x)∑y∈γ

a+(x− y), x ∈ Rd \ γ, γ ∈ Γ, (3.69)

where 0 < m ∈ L∞(Rd), 0 ≤ κ± ∈ L∞(Rd), 0 ≤ a± ∈ L1(Rd, dx)∩L∞(Rd, dx),∫Rd a

±(x)dx = 1. Then, by (3.17), (3.20), and (2.18)–(2.19),

Id(ξ) =∑x∈ξ

d(x, ξ \ x) +∑x∈ξ

Cκ−(x), Ib(ξ) =∑x∈ξ

b(x, ξ \ x) +∑x∈ξ

Cκ+(x).

Let us suppose, cf. [26], that there exists δ > 0 such that

(4 + δ)Cκ−(x) ≤ m(x), x ∈ Rd, (3.70)

(4 + δ)κ+(x) ≤ m(x), x ∈ Rd, (3.71)

4κ+(x)a+(x) ≤ Cκ−(x)a−(x). x ∈ Rd, (3.72)

Then

d(x, ξ) + Cκ−(x) ≤ d(x, ξ) +m(x)

4 + δ≤(

1 +1

4 + δ

)d(x, ξ),

b(x, ξ) + Cκ+(x) ≤ C

4κ−(x)

∑y∈ξ

a−(x− y) +Cm(x)

4 + δ<C

4d(x, ξ),

Hence, (3.30), (3.31) hold with

a1 = 1 +1

4 + δ, a2 =

C

4,

that fulfills (3.32). Next, under conditions (3.70), (3.72), we have

d(x, ξ) ≤ ‖m‖L∞(Rd) + ‖κ−‖L∞(Rd)‖a−‖L∞(Rd)|ξ|, ξ ∈ Γ0,

and hence (3.45) holds with ν = 1, which makes (3.51) obvious.

Remark 3.20. It was shown in [26] that, for the case of constant m,κ±, thecondition like (3.70) is essential. Namely, if m > 0 is arbitrary small the operator

L will not be even accretive in LC .

Example 3.21. (Contact model with establishment). Let L be given by (3.1)with d(x, γ) = m(x) for all γ ∈ Γ and

b(x, γ) = κ(x) exp{∑y∈γ

φ(x− y)}∑y∈γ

a(x− y), γ ∈ Γ, x ∈ Rd \ γ. (3.73)

Here 0 < m ∈ L∞(Rd), 0 ≤ κ ∈ L∞(Rd), 0 ≤ a ∈ L1(Rd) ∩ L∞(Rd),∫Rd a(x) dx = 1.

34

3.6 Stationary equation

In this subsection we study the question about stationary solutions to (2.41).For any s ≥ 0, we consider the following subset of KC

K(s)αC :=

{k ∈ KαC

∣∣ k(∅) = s}.

We define K to be the closure of K(0)αC in the norm of KC . It is clear that K with

the norm of KC is a Banach space.

Proposition 3.22. Let (3.30), (3.31), and (3.45) be satisfied with

a1 +a2

C< 2. (3.74)

Assume, additionally, that

d(x, ∅) > 0, x ∈ Rd. (3.75)

Then for any α ∈ (0; 1ν ) the stationary equation

L∗k = 0 (3.76)

has a unique solution kinv from K(1)αC which is given by the expression

kinv = 1∗ + (11− S)−1E. (3.77)

Here 1∗ denotes the function defined by 1∗(η) = 0|η|, η ∈ Γ0, the function

E ∈ K(0)αC is such that

E(η) = 11Γ(1)(η)∑x∈η

b(x, ∅)d(x, ∅)

, η ∈ Γ0,

and S is a generalized Kirkwood–Salzburg operator on K, given by

(Sk) (η) =− 1

D (η)

∑x∈η

∫Γ0\{∅}

k(ζ ∪ η)(K−1d(x, · ∪ η \ x))(ζ)dλ(ζ) (3.78)

+1

D (η)

∑x∈η

∫Γ0

k(ζ ∪ (η \ x))(K−1b(x, · ∪ η \ x))(ζ)dλ(ζ),

for η 6= ∅ and (Sk) (∅) = 0. In particular, if b(x, ∅) = 0 for a.a. x ∈ Rd thenthis solution is such that

k(n)inv = 0, n ≥ 1. (3.79)

Remark 3.23. It is worth noting that (3.41), (3.42) imply (3.75).

35

Proof. Suppose that (3.76) holds for some k ∈ K(1)αC . Then

D (η) k(η) =−∑x∈η

∫Γ0\{∅}

k(ζ ∪ η)(K−1d(x, · ∪ η \ x)

)(ζ)dλ(ζ)

+∑x∈η

∫Γ0

k(ζ ∪ (η \ x))(K−1b(x, · ∪ η \ x)

)(ζ)dλ(ζ). (3.80)

The equality (3.80) is satisfied for any k ∈ K(1)αC at the point η = ∅. Using the fact

that D(∅) = 0 one may rewrite (3.80) in terms of the function k = k−1∗ ∈ K(0)αC .

Namely,

D (η) k(η) =−∑x∈η

∫Γ0\{∅}

k(ζ ∪ η)(K−1d(x, · ∪ η \ x)

)(ζ)dλ(ζ)

+∑x∈η

∫Γ0

k(ζ ∪ (η \ x))(K−1b(x, · ∪ η \ x)

)(ζ)dλ(ζ).

+∑x∈η

0|η\x|b(x, η \ x). (3.81)

As a result,k(η) = (Sk)(η) + E(η), η ∈ Γ0.

Next, for η 6= ∅

C−|η| |(Sk) (η)|

≤C−|η|

D (η)

∑x∈η

∫Γ0\{∅}

|k(ζ ∪ η)|∣∣(K−1d(x, · ∪ η \ x))(ζ)

∣∣ dλ(ζ)

+C−|η|

D (η)

∑x∈η

∫Γ0

k(ζ ∪ (η \ x))∣∣(K−1b(x, · ∪ η \ x))(ζ)

∣∣ dλ(ζ)

≤‖k‖KCD (η)

∑x∈η

∫Γ0\{∅}

C |ζ|∣∣(K−1d(x, · ∪ η \ x))(ζ)

∣∣ dλ(ζ)

+‖k‖KCD (η)

1

C

∑x∈η

∫Γ0

C |ζ|∣∣(K−1b(x, · ∪ η \ x))(ζ)

∣∣ dλ(ζ)

≤‖k‖KCD (η)

D (η)(a1 − 1 +

a2

C

)=(a1 − 1 +

a2

C

)‖k‖KC .

Hence,

‖S‖ = a1 +a2

C− 1 < 1

in ‹K. This finishes the proof.

36

Remark 3.24. The name of the operator (3.78) is motivated by Example 3.18.Namely, if s = 0 then the operator (3.78) has form

(Sk) (η) =1

m|η|∑x∈η

eλ(e−φ(x−·), η \ x)

∫Γ0

k(ζ ∪ (η \ x))eλ(e−φ(x−·) − 1, ζ)dλ(ζ),

that is quite similar of the so-called Kirkwood–Salsburg operator known inmathematical physics (see e.g. [49, 75]). For s = 0 condition (3.74) has formzC e

Cβ−1 < 1. Under this condition, the stationary solution to (3.76) is a uniqueand coincides with the correlation function of the Gibbs measure, correspondingto potential φ and activity z.

Remark 3.25. It is worth pointing out that b(x, ∅) = 0 in the case of Exam-ple 3.19. Therefore, if we suppose (cf. (3.70), (3.72)) that 2κ−C < m and2κ+a+(x) ≤ Cκ−a−(x), for x ∈ Rd, condition (3.74) will be satisfied. How-ever, the unique solution to (3.76) will be given by (3.79). In the next examplewe improve this statement.

Example 3.26. Let us consider the following natural modification of BDLP-model coming from Example 3.19: let d be given by (3.68) and

b(x, γ) = κ+ κ+∑y∈γ

a+(x− y), x ∈ Rd \ γ, γ ∈ Γ, (3.82)

where κ+, a+ are as before and κ > 0. Then, under assumptions

2 max{κ−C;

2κ

C

}< m (3.83)

and2κ+a+(x) ≤ Cκ−a−(x), x ∈ Rd, (3.84)

we obtain for some δ > 0∫Γ0

∣∣K−1d (x, · ∪ ξ)∣∣ (η)C |η|dλ (η) = d(x, ξ) + Cκ− ≤

(1 +

1

2 + δ

)d(x, ξ)∫

Γ0

∣∣K−1b (x, · ∪ ξ)∣∣ (η)C |η|dλ (η) = b(x, ξ) + Cκ+

≤ κ+1

2Cκ−

∑y∈ξ

a−(x− y) +m

4C <

C

2d(x, ξ).

The latter inequalities imply (3.74). In this case, E(η) = 11Γ(1)(η) κm .

Remark 3.27. If a+(x) = a−(x), x ∈ Rd and κ+ = zκ−, κ = zm for some z > 0then b(x, γ) = zd(x, γ) and the Poisson measure πz with the intensity z willbe symmetrizing measure for the operator L. In particular, it will be invariantmeasure. This fact means that its correlation function kz(η) = z|η| is a solutionto (3.76). Conditions (3.83) and (3.84) in this case are equivalent to 4z < Cand 2κ−C < m. As a result, due to uniqueness of such solution,

1∗(η) + z(11− S)−111Γ(1)(η) = z|η|, η ∈ Γ0.

37

4 Approximative approach for the Glauber dy-namics

In this section we consider an approximative approach for the construction ofthe Glauber-type dynamics described in Example 3.18 for

s = 0, m(x) ≡ 1, z(x) ≡ z > 0.

Therefore, in such a case, (3.1) has the form

(LF )(γ) :=∑x∈γ

[F (γ \ x)− F (γ)

](4.1)

+ z

∫Rd

[F (γ ∪ x)− F (γ)

]exp{−Eφ(x, γ)

}dx, γ ∈ Γ,

with Eφ given by (2.11).Let G ∈ Bbs(Γ0) then F = KG ∈ Fcyl(Γ). By (3.6), (3.17), (3.22), one has

the following explicit form for the mapping L := K−1LK on Bbs(Γ0)

(LG)(η) = −|η|G(η)

+ z∑ξ⊂η

∫Rde−E

φ(x,ξ)G(ξ ∪ x)eλ(e−φ(x−·) − 1, η \ ξ)dx, (4.2)

where eλ is given by (2.25).Let us denote, for any η ∈ Γ0,

(L0G)(η) := −|η|G(η); (4.3)

(L1G)(η) := z∑ξ⊂η

∫Rde−E

φ(x,ξ)G(ξ ∪ x)eλ(e−φ(x−·) − 1, η \ ξ)dx. (4.4)

To simplify notation we continue to write Cφ for β−1. In contrast to (3.29),we will not work the maximal domain of the operator L0. Namely, the followingstatement will be used

Proposition 4.1. The expression (4.2) defines a linear operator L in LC withthe dense domain L2C ⊂ LC .

Proof. For any G ∈ L2C

‖L0G‖C =

∫Γ0

|G(η)||η|C |η|dλ(η) <

∫Γ0

|G(η)|2|η|C |η|dλ(η) <∞

38

and, by Lemma 3.4,

‖L1G‖C

≤ z∫

Γ0

∑ξ⊂η

∫Rd

e−Eφ(x,ξ) |G(ξ ∪ x)| eλ

(∣∣∣e−φ(x−·) − 1∣∣∣ , η \ ξ) dxC |η|dλ (η)

= z

∫Γ0

∫Γ0

∫Rd

e−Eφ(x,ξ) |G(ξ ∪ x)| eλ

(∣∣∣e−φ(x−·) − 1∣∣∣ , η) dxC |η|C |ξ|dλ (ξ) dλ (η)

≤ z

Cexp {CCφ}

∫Γ0

|G (ξ)| |ξ|C |ξ|dλ (ξ) <z

Cexp {CCφ}

∫Γ0

|G (ξ)| 2|ξ|C |ξ|dλ (ξ)

<∞.

Embedding L2C ⊂ LC is dense since Bbs(Γ0) ⊂ L2C .

4.1 Description of approximation

In this section we will use the symbol K0 to denote the restriction of K ontofunctions on Γ0.

Let δ ∈ (0; 1) be arbitrary and fixed. Consider for any Λ ∈ Bb(Rd) thefollowing linear mapping on functions F ∈ K0

(Bbs(Γ0)

)⊂ Fcyl(Γ)(

PΛδ F)

(γ) =∑η⊂γ

δ|η| (1− δ)|γ\η|(ΞΛδ (γ)

)−1(4.5)

×∫

ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,γ)F ((γ \ η) ∪ ω) dλ (ω) , γ ∈ Γ0,

where

ΞΛδ (γ) =

∫ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,γ)dλ (ω) . (4.6)

Clearly, PΛδ is a positive preserving mapping and(

PΛδ 1)

(γ) =∑η⊂γ

δ|η| (1− δ)|γ\η| = 1, γ ∈ Γ0.

Operator (4.5) is constructed as a transition operator of a Markov chain,which is a time discretization of a continuous time process with the generator(4.1) and discretization parameter δ ∈ (0; 1). Roughly speaking, according tothe representation (4.5), the probability of transition γ → (γ \ η)∪ω (which de-scribes removing of subconfiguration η ⊂ γ and birth of a new subconfigurationω ∈ Γ(Λ)) after small time δ is equal to(

ΞΛδ (γ)

)−1δ|η|(1− δ)|γ\η|(zδ)|ω|

∏y∈ω

e−Eφ(y,γ).

We may rewrite (4.5) in another manner.

39

Proposition 4.2. For any F ∈ Fcyl(Γ0) the following equality holds(PΛδ F)

(γ) =∑ξ⊂γ

(1− δ)|ξ|∫

ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,γ) (4.7)

× (K−10 F ) (ξ ∪ ω) dλ (ω) .

Proof. Let G := K−10 F ∈ Bbs(Γ0). Since ΞΛ

δ doesn’t depend on η, for γ ∈ Γ0

we have (PΛδ F)

(γ) =(ΞΛδ (γ)

)−1∫

ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,γ) (4.8)

×∑η⊂γ

δ|γ\η| (1− δ)|η| F (η ∪ ω) dλ (ω) .

To rewrite (4.5), we have used also that any η ⊂ γ corresponds to a uniqueγ \ η ⊂ γ. Applying the definition of K0 to F = K0G we obtain∑η⊂γ

δ|γ\η| (1− δ)|η| F (η ∪ ω) =∑η⊂γ

δ|γ\η| (1− δ)|η|∑ζ⊂η

∑β⊂ω

G (ζ ∪ β) (4.9)

=∑ζ⊂γ

∑β⊂ω

G (ζ ∪ β)∑

η′⊂γ\ζ

δ|γ\(η′∪ζ)| (1− δ)|η

′∪ζ| ,

where after changing summation over η ⊂ γ and ζ ⊂ η we have used the factthat for any configuration η ⊂ γ which contains fixed ζ ⊂ γ there exists a uniqueη′ ⊂ γ \ ζ such that η = η′ ∪ ζ. But by the binomial formula∑

η′⊂γ\ζ

δ|γ\(η′∪ζ)| (1− δ)|η

′∪ζ| = (1− δ)|ζ|∑

η′⊂γ\ζ

δ|(γ\ζ)\η′| (1− δ)|η

′| (4.10)

= (1− δ)|ζ|(δ + 1− δ)|γ\ζ| = (1− δ)|ζ|.

Combining (4.8), (4.9), (4.10), we get(PΛδ F)

(γ) =(ΞΛδ (γ)

)−1∫

ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,γ)

×∑ζ⊂γ

∑β⊂ω

G (ζ ∪ β) (1− δ)|ζ|dλ (ω) .

Next, Lemma 3.4 yields(PΛδ F)

(γ) =(ΞΛδ (γ)

)−1∫

ΓΛ

∫ΓΛ

(zδ)|ω∪β| ∏

y∈ω∪β

e−Eφ(y,γ)

×∑ζ⊂γ

G (ζ ∪ β) (1− δ)|ζ|dλ (ω) dλ (β)

=

∫ΓΛ

(zδ)|β|∏

y∈β

e−Eφ(y,γ)

∑ζ⊂γ

G (ζ ∪ β) (1− δ)|ζ|dλ (β) ,

which proves the statement.

40

In the next proposition we describe the image of PΛδ under the K0-transform.

Proposition 4.3. Let “PΛδ = K−1

0 PΛδ K0. Then for any G ∈ Bbs(Γ0) the fol-

lowing equality holds(“PΛδ G)

(η) =∑ξ⊂η

(1− δ)|ξ|∫

ΓΛ

(zδ)|ω|G (ξ ∪ ω) (4.11)

×∏y∈ξ

e−Eφ(y,ω)

∏y′∈η\ξ

(e−E

φ(y′,ω) − 1)dλ (ω) , η ∈ Γ0.

Proof. By (4.7) and the definition of K−10 , we have(“PΛ

δ G)

(η)

=∑ζ⊂η

(−1)|η\ζ|∑ξ⊂ζ

(1− δ)|ξ|∫

ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,ζ)G (ξ ∪ ω) dλ (ω)

=∑ξ⊂η

(1− δ)|ξ|∑ζ⊂η\ξ

(−1)|(η\ξ)\ζ|∫

ΓΛ

(zδ)|ω|∏

y∈ωe−E

φ(y,ζ∪ξ)G (ξ ∪ ω) dλ (ω) .

By the definition of the relative energy∏y∈ω

e−Eφ(y,ζ∪ξ) =

∏y∈ξ

e−Eφ(y,ω)

∏y′∈ζ

e−Eφ(y′,ω).

The well-known equality (see, e.g., [36])∑ζ⊂η\ξ

(−1)|(η\ξ)\ζ|∏y′∈ζ

e−Eφ(y′,ω) =

(K−1

0

∏y′∈·

e−Eφ(y′,ω)

)(η \ ξ)

=∏

y′∈η\ξ

(e−E

φ(y′,ω) − 1)

completes the proof.

4.2 Construction of the semigroup on LC

By analogy with (4.11), we consider the following linear mapping on measurablefunctions on Γ0(“PδG) (η) :=

∑ξ⊂η

(1− δ)|ξ|∫

Γ0

(zδ)|ω|G (ξ ∪ ω) (4.12)

×∏y∈ξ

e−Eφ(y,ω)

∏y′∈η\ξ

(e−E

φ(y′,ω) − 1)dλ (ω) , η ∈ Γ0.

Proposition 4.4. LetzeCCφ ≤ C. (4.13)

Then “Pδ, given by (4.12), is a well defined linear operator in LC , such that∥∥“Pδ∥∥ ≤ 1. (4.14)

41

Proof. Since φ ≥ 0 we have∥∥∥“PδG∥∥∥C≤∫

Γ0

∑ξ⊂η

(1− δ)|ξ|∫

Γ0

(zδ)|ω| |G (ξ ∪ ω)|

×∏y∈ξ

e−Eφ(y,ω)

∏y′∈η\ξ

∣∣∣e−Eφ(y′,ω) − 1∣∣∣ dλ (ω)C |η|dλ (η)

=

∫Γ0

∫Γ0

(1− δ)|ξ|∫

Γ0

(zδ)|ω| |G (ξ ∪ ω)|

×∏y∈ξ

e−Eφ(y,ω)

∏y′∈η

∣∣∣e−Eφ(y′,ω) − 1∣∣∣ dλ (ω)C |η|C |ξ|dλ (ξ) dλ (η)

=

∫Γ0

∫Γ0

(1− δ)|ξ| (zδ)|ω| |G (ξ ∪ ω)|

×∏y∈ξ

e−Eφ(y,ω) exp

ßC

∫Rd

(1− e−E

φ(y′,ω))dy′™dλ (ω)C |ξ|dλ (ξ) .

It is easy to see by the induction principle that for φ ≥ 0, ω ∈ Γ0, y /∈ ω

1− e−Eφ(y,ω) = 1−

∏x∈ω

e−φ(x−y) ≤∑x∈ω

Ä1− e−φ(x−y)

ä. (4.15)

Then∥∥“PδG∥∥C ≤ ∫Γ0

∫Γ0

(1− δ)|ξ| (zδ)|ω| |G (ξ ∪ ω)|

× exp

{C∑x∈ω

∫Rd

Ä1− e−φ(x−y)

ädy

}dλ (ω)C |ξ|dλ (ξ)

=

∫Γ0

∫Γ0

(1− δ)|ξ| (zδ)|ω| |G (ξ ∪ ω)| eCCφ|ω|C |ξ|dλ (ω) dλ (ξ)

=

∫Γ0

[(1− δ)C + zδeCCφ

]|ω| |G (ω)| dλ (ω) ≤ ‖G‖C .

For the last inequality we have used that (4.13) implies (1− δ)C+zδeCCφ ≤ C.Note that, for λ-a.a. η ∈ Γ0 (“PδG)(η) <∞, (4.16)

and the statement is proved.

Proposition 4.5. Let the inequality (4.13) be fulfilled and define

Lδ :=1

δ(“Pδ − 11), δ ∈ (0; 1),

where 11 is the identity operator in LC . Then for any G ∈ L2C∥∥(Lδ − L)G∥∥C≤ 3δ‖G‖2C . (4.17)

42

Proof. Let us denote(“P (0)δ G

)(η) =

∑ξ⊂η

(1− δ)|ξ|G (ξ) 0|η\ξ| = (1− δ)|η|G (η) ; (4.18)

(“P (1)δ G

)(η) = zδ

∑ξ⊂η

(1− δ)|ξ|∫RdG (ξ ∪ x) (4.19)

×∏y∈ξ

e−φ(y−x)∏y∈η\ξ

Äe−φ(y−x) − 1

ädx; (4.20)

and “P (≥2)δ = “Pδ − Ä“P (0)

δ + “P (1)δ

ä. (4.21)

Clearly∥∥(Lδ − L)G∥∥C

=

∥∥∥∥1

δ

Ä“PδG−Gä− LG∥∥∥∥C

(4.22)

≤∥∥∥∥1

δ

Ä“P (0)δ G−G

ä− L0G

∥∥∥∥C

+

∥∥∥∥1

δ“P (1)δ G− L1G

∥∥∥∥C

+1

δ

∥∥∥“P (≥2)δ G

∥∥∥C.

Now we estimate each of the terms in (4.22) separately. By (4.3) and (4.18), wehave∥∥∥∥1

δ

Ä“P (0)δ G−G

ä− L0G

∥∥∥∥C

=

∫Γ0

∣∣∣∣∣ (1− δ)|η| − 1

δ+ |η|

∣∣∣∣∣ |G (η)|C |η|dλ (η) .

But, for any |η| ≥ 2∣∣∣∣∣ (1− δ)|η| − 1

δ+ |η|

∣∣∣∣∣ =

∣∣∣∣∣∣|η|∑k=2

Ç|η|k

å(−1)kδk−1

∣∣∣∣∣∣= δ

∣∣∣∣∣∣|η|∑k=2

Ç|η|k

å(−1)kδk−2

∣∣∣∣∣∣ ≤ δ|η|∑k=2

Ç|η|k

å< δ · 2|η|.

Therefore, ∥∥∥∥1

δ

Ä“P (0)δ G−G

ä− L0G

∥∥∥∥C

≤ δ‖G‖2C . (4.23)

Next, by (4.4) and (4.20), one can write∥∥∥∥1

δ“P (1)δ G− L1G

∥∥∥∥C

= z

∫Γ0

∣∣∣∣∑ξ⊂η

Ä(1− δ)|ξ| − 1

ä∫RdG (ξ ∪ x)

∏y∈ξ

e−φ(y−x)

×∏y∈η\ξ

Äe−φ(y−x) − 1

ädx

∣∣∣∣C |η|dλ (η)

≤ z∫

Γ0

∫Γ0

Ä1− (1− δ)|ξ|

ä∫Rd|G (ξ ∪ x)|

∏y∈ξ

e−φ(y−x)

×∏y∈η

Ä1− e−φ(y−x)

ädxC |ξ|C |η|dλ (ξ) dλ (η) ,

43

where we have used Lemma 3.4. Note that for any |ξ| ≥ 1

1− (1− δ)|ξ| = δ

|ξ|−1∑k=0

(1− δ)k ≤ δ |ξ|

Then, by (4.13) and (2.12), one may estimate∥∥∥∥1

δ“P (1)δ G− L1G

∥∥∥∥C

≤ zδ∫

Γ0

|ξ|∫Rd|G (ξ ∪ x)| dxC |ξ|eCCφdλ (ξ) (4.24)

≤ zδ∫

Γ0

|ξ| (|ξ| − 1) |G (ξ)|C |ξ|−1eCCφdλ (ξ) .

Since n (n− 1) ≤ 2n, n ≥ 1 and by (4.13), the latter expression can be boundedby

δ

∫Γ0

|G (ξ)| (2C)|ξ|λ (dξ) .

Finally, Lemma 3.4, (4.15) and bound e−Eφ(y,ω) ≤ 1, imply (we set here

Γ(≥2)0 :=

⊔n≥2 Γ(n))∥∥∥∥1

δ“P (≥2)δ G

∥∥∥∥C

≤ 1

δ

∫Γ0

∑ξ⊂η

(1− δ)|ξ|∫

Γ(≥2)0

(zδ)|ω| |G (ξ ∪ ω)| (4.25)

×∏y∈ξ

e−Eφ(y,ω)

∏y∈η\ξ

Ä1− e−E

φ(y,ω)ädλ (ω)C |η|dλ (η)

≤ δ∫

Γ0

∑ξ⊂η

(1− δ)|ξ|∫

Γ(≥2)0

z|ω| |G (ξ ∪ ω)|

×∏y∈ξ

e−Eφ(y,ω)

∏y∈η\ξ

Ä1− e−E

φ(y,ω)ädλ (ω)C |η|dλ (η)

≤ δ∫

Γ0

∑ξ⊂η

(1− δ)|ξ|∫

Γ0

z|ω| |G (ξ ∪ ω)|

×∏y∈ξ

e−Eφ(y,ω)

∏y∈η\ξ

Ä1− e−E

φ(y,ω)ädλ (ω)C |η|dλ (η)

≤ δ∫

Γ0

∫Γ0

(1− δ)|ξ| z|ω| |G (ξ ∪ ω)|

×∫

Γ0

∏y∈η

Ä1− e−E

φ(y,ω)äC |η|dλ (η) dλ (ω)C |ξ|dλ (ξ)

≤ δ∫

Γ0

∫Γ0

(1− δ)|ξ| z|ω| |G (ξ ∪ ω)| eCCφ|ω|dλ (ω)C |ξ|dλ (ξ)

≤ δ∫

Γ0

[(1− δ)C + zeCCφ

]|ω| |G (ω)| dλ (ω)

≤ δ∫

Γ0

[(2− δ)C]|ω| |G (ω)| dλ (ω) ≤ δ

∫Γ0

|G (ω)| (2C)|ω|dλ (ω) .

44

Combining inequalities (4.23)–(4.25) we obtain the assertion of the proposition.

We will need the following results in the sequel.

Lemma 4.6 (cf. [20, Corollary 3.8]). Let A be a linear operator on a Banachspace L with D (A) dense in L, and let ||| · ||| be a norm on D (A) with respectto which D (A) is a Banach space. For n ∈ N let Tn be a linear ‖·‖-contractionon L such that Tn : D (A)→ D (A), and define An = n (Tn − 1). Suppose thereexist ω ≥ 0 and a sequence {εn} ⊂ (0; +∞) tending to zero such that for n ∈ N

‖(An −A) f‖ ≤ εn|||f |||, f ∈ D (A) (4.26)

and ∣∣∣∣∣∣Tn �D(A)

∣∣∣∣∣∣ ≤ 1 +ω

n. (4.27)

Then A is closable and the closure of A generates a strongly continuous con-traction semigroup on L.

Lemma 4.7 (cf. [20, Theorem 6.5]). Let L,Ln, n ∈ N be Banach spaces, andpn : L → Ln be bounded linear transformation, such that supn ‖pn‖ < ∞.For any n ∈ N, let Tn be a linear contraction on Ln, let εn > 0 be such thatlimn→∞ εn = 0, and put An = ε−1

n (Tn − 11). Let T (t) be a strongly continuouscontraction semigroup on L with generator A and let D be a core for A. Thenthe following are equivalent:

1. For each f ∈ L, T[t/εn]n pnf → pnT (t)f in Ln for all t ≥ 0 uniformly

on bounded intervals. Here and below [ · ] mean the entire part of a realnumber.

2. For each f ∈ D, there exists fn ∈ Ln for each n ∈ N such that fn → pnfand Anfn → pnAf in Ln.

And now we are able to show the existence of the semigroup on LC .

Theorem 4.8. Let

z ≤ min{Ce−CCφ ; 2Ce−2CCφ

}. (4.28)

Then(L,L2C

)from Proposition 4.1 is a closable linear operator in LC and its

closure(L,D(L)

)generates a strongly continuous contraction semigroup Tt on

LC .

Proof. We apply Lemma 4.6 for L = LC ,(A,D(A)

)=(L,L2C

), ||| · ||| := ‖ · ‖2C ;

Tn = “Pδ and An = n (Tn − 1) = 1δ (“Pδ − 11) = Lδ, where δ = 1

n , n ≥ 2.Condition zeCCφ ≤ C, Proposition 4.4, and Proposition 4.5 provide that Tn,

n ≥ 2 are linear ‖ · ‖C-contractions and (4.26) holds with εn = 3n = 3δ. On the

other hand, in addition, Proposition 4.4 applied to the constant 2C instead ofC gives (4.27) for ω = 0 under condition ze2CCφ ≤ 2C.

45

Moreover, since we proved the existence of the semigroup Tt on LC one canapply contractions “Pδ defined above by (4.12) to approximate the semigroup Tt.

Corollary 4.9. Let (4.13) holds. Then for any G ∈ LC(“P 1n

)[nt]G→ TtG, n→∞

for all t ≥ 0 uniformly on bounded intervals.

Proof. The statement is a direct consequence of Theorem 4.8, convergence(4.17), and Lemma 4.7 (if we set Ln = L = LC , pn = 11, n ∈ N).

4.3 Finite-volume approximation of “Tt

Note that “Pδ defined by (4.12) is a formal point-wise limit of “PΛδ as Λ ↑ Rd.

We have shown in (4.16) that this definition is correct. Corollary 4.9 claims

additionally that the linear contractions “Pδ approximate the semigroup Tt, whenδ ↓ 0. One may also show that mappings “PΛ

δ have a similar property whenΛ ↑ Rd, δ ↓ 0.

Let us fix a system {Λn}n≥2, where Λn ∈ Bb(Rd), Λn ⊂ Λn+1,⋃n Λn = Rd.

We setTn := “PΛn

1n

.

Note that any Tn is a linear mapping on Bbs(Γ0). We consider also the systemof Banach spaces of measurable functions on Γ0

LC,n :=

ßG : Γ(Λn)→ R

∣∣∣∣ ‖G‖C,n :=

∫Γ(Λn)

|G(η)|C |η|dλ(η) <∞™.

Let pn : LC → LC,n be a cut-off mapping, namely, for any G ∈ LC

(pnG)(η) = 11Γ(Λn)(η)G(η).

Then, obviously, ‖pnG‖C,n ≤ ‖G‖C . Hence, pn : LC → LC,n is a linear boundedtransformation with ‖pn‖ = 1.

Proposition 4.10. Let (4.13) hold. Then for any G ∈ LC∥∥(Tn)[nt]pnG− pnTtG∥∥C,n → 0, n→∞

for all t ≥ 0 uniformly on bounded intervals.

Proof. The proof of the proposition is completed by showing that all conditionsof Lemma 4.7 hold. Using completely the same arguments as in the proof ofProposition 4.4 one gets that each Tn = “PΛn

1n

is a linear contraction on LC,n,

n ≥ 2 (note that for any n ≥ 2, (2.12) implies∫

Λn

(1 − e−φ(x)

)dx ≤ Cφ < ∞).

Next, we set An = n(Tn − 11n) where 11n is a unit operator on LC,n and letus expand Tn in three parts analogously to the proof of Proposition 4.5: Tn =

46

T(0)n + T

(1)n + T

(≥2)n . As a result, An = n(T

(0)n − 11n) + nT

(1)n + nT

(≥2)n . For any

G ∈ L2C we set Gn = pnG ∈ L2C,n ⊂ LC,n. To finish the proof we have toverify that for any G ∈ L2C

‖AnGn − pnLG‖C,n → 0, n→∞. (4.29)

For any G ∈ L2C

‖AnGn − pnLG‖C,n ≤‖n(T (0)n − 11n)Gn − pnL0G‖C,n (4.30)

+ ‖nT (1)n Gn − pnL1G‖C,n + ‖nT (≥2)

n Gn‖C,n.

Note, that pnL0G = L0Gn. Using the same arguments as in the proof ofProposition 4.5 we obtain

‖n(T (0)n − 11n)Gn − pnL0G‖C,n + ‖nT (≥2)

n Gn‖C,n ≤2

n‖G‖2C,n ≤

2

n‖G‖2C .

Next,

‖nT (1)n Gn − pnL1G‖C,n

≤ z∫

ΓΛn

∑ξ⊂η

∫Rd

∣∣∣∣∣Å

1− 1

n

ã|ξ|11Λn(x)− 1

∣∣∣∣∣ |G (ξ ∪ x) |

×∏y∈ξ

e−φ(y−x)∏y∈η\ξ

Ä1− e−φ(y−x)

ädxC |η|dλ (η)

≤ z∫

Γ(Λn)

∫Γ(Λn)

∫Rd

ñ1−Å

1− 1

n

ã|ξ|11Λn(x)

ô|G (ξ ∪ x) |

×∏y∈η

Ä1− e−φ(y−x)

ädxC |η∪ξ|dλ (η) dλ (ξ)

≤C∫

Γ(Λn)

∫Rd

ñ1−Å

1− 1

n

ã|ξ|11Λn(x)

ô|G (ξ ∪ x) |dxC |ξ|dλ (ξ) ,

where we have used (2.12) and (4.13). Using the same estimates as for (4.24)we may continue

≤C∫

Γ(Λn)

∫Λn

ñ1−Å

1− 1

n

ã|ξ|ô|G (ξ ∪ x) |dxC |ξ|dλ (ξ)

+ C

∫Γ(Λn)

∫Λcn

|G (ξ ∪ x) |dxC |ξ|dλ (ξ)

≤ 1

n‖G‖2C,n + C

∫Γ0

∫Λcn

|G (ξ ∪ x) |dxC |ξ|dλ (ξ) .

But by the Lebesgue dominated convergence theorem,∫Γ0

∫Λcn

|G (ξ ∪ x) |dxC |ξ|dλ (ξ)→ 0, n→∞.

47

Indeed, 11Λcn(x)|G (ξ ∪ x) | → 0 point-wisely and may be estimated on Γ0 × Rd

by |G (ξ ∪ x) | which is integrable:

C

∫Γ0

∫Rd|G (ξ ∪ x) |dxC |ξ|dλ (ξ) =

∫Γ0

|ξ||G(ξ)|C |ξ|dλ (ξ) ≤ ‖G‖2C <∞.

Therefore, by (4.30), the convergence (4.29) holds for any G ∈ L2C , whichcompletes the proof.

4.4 Evolution of correlation functions

Under condition (4.28), we proceed now to the same arguments as in Subsec-

tion 3.4. Namely, one can construct the restriction T�(t) of the semigroup of

T ∗(t) onto the Banach space D(L∗) (recall that the closure is in the norm of

KC). Note that the domain of the dual operator to (L,L2C) might be bigger

than the domain considered in Subsection 3.4. Nevertheless, T�(t) will be a C0-

semigroup on D(L∗) and its generator L� will be a part of L∗, namely, (3.44)

holds and L∗k = L�k for any k ∈ D(L�).The next statement is a straightforward consequence of Proposition 3.12.

Proposition 4.11. For any α ∈ (0; 1) the following inclusions hold KαC ⊂D(L∗) ⊂ D(L∗) ⊂ KC .

Then, by Proposition 3.5, we immediately obtain that, for k ∈ KαC ,

(L∗k)(η) =− |η|k(η) (4.31)

+ z∑x∈η

e−Eφ(x,η\x)

∫Γ0

eλ(e−φ(x−·) − 1, ξ)k((η \ x) ∪ ξ) dλ(ξ).

The next statement is an analog of Proposition 3.15.

Proposition 4.12. Suppose that (4.28) is satisfied. Furthermore, we addition-ally assume that

z < Ce−CCφ , if CCφ ≤ ln 2. (4.32)

Then there exists α0 = α0(z, φ, C) ∈ (0; 1) such that for any α ∈ (α0; 1) the set

KαC is the T ∗(t)-invariant linear subspace of KC .

Proof. Let us consider function f(x) := xe−x, x ≥ 0. It has the following prop-erties: f is increasing on [0; 1] from 0 to e−1 and it is asymptotically decreasingon [1; +∞) from e−1 to 0; f(x) < f(2x) for x ∈ (0, ln 2); x = ln 2 is the onlynon-zero solution to f(x) = f(2x).

By assumption (4.28), zCφ ≤ min{CCφe−CCφ , 2CCφe−2CCφ}. Therefore, ifCCφe

−CCφ 6= 2CCφe−2CCφ then (4.28) with necessity implies

zCφ < e−1. (4.33)

48

This inequality remains also true if CCφ = ln 2 because of (4.32). Under condi-tion (4.33), the equation f(x) = zCφ has exactly two roots, say, 0 < x1 < 1 <x2 < +∞. Then, (4.32) implies x1 < CCφ < 2CCφ ≤ x2.

If CCφ > 1 then we set α0 := max¶

12 ; 1

CCφ; 1C

©< 1. This yields 2αCCφ >

CCφ and αCCφ > 1 > x1. If x1 < CCφ ≤ 1 then we set α0 := max¶

12 ; x1

CCφ; 1C

©<

1 that gives 2αCCφ > CCφ and αCCφ > x1.As a result,

x1 < αCCφ < CCφ < 2αCCφ < 2CCφ ≤ x2 (4.34)

and 1 < αC < C < 2αC < 2C. The last inequality shows that L2C ⊂ L2αC ⊂LC ⊂ LαC . Moreover, by (4.34), we may prove that the operator (L,L2αC) is

closable in LαC and its closure is a generator of a contraction semigroup Tα(t)on LαC . The proof is identical to the proofs above.

It is easy to see, that Tα(t)G = T (t)G for any G ∈ LC . Indeed, from the

construction of the semigroup T (t) and analogous construction for the semigroup

Tα(t), we have that there exists family of mappings “Pδ, δ > 0 independent of

α and C, given by (4.12), such that “P [ tδ ]δ for any t ≥ 0 strongly converges to

T (t) and Tα(t) in LC and LαC , correspondingly, as δ → 0. Here and below [ · ]means the entire part of a number. Then for any G ∈ LC ⊂ LαC we have thatT (t)G ∈ LC ⊂ LαC and Tα(t)G ∈ LαC and

‖T (t)G− Tα(t)G‖αC ≤∥∥∥T (t)G− “P [ tδ ]

δ G∥∥∥αC

+∥∥∥Tα(t)G− “P [ tδ ]

δ G∥∥∥αC

≤∥∥∥T (t)G− “P [ tδ ]

δ G∥∥∥C

+∥∥∥Tα(t)G− “P [ tδ ]

δ G∥∥∥αC→ 0,

as δ → 0. Therefore, T (t)G = Tα(t)G in LαC (recall that G ∈ LC) that yields

T (t)G(η) = Tα(t)G(η) for λ-a.a. η ∈ Γ0 and, therefore, T (t)G = Tα(t)G in LC .Note that for any G ∈ LC ⊂ LαC and for any k ∈ KαC ⊂ KC we have

Tα(t)G ∈ LαC and ¨Tα(t)G, k

∂∂=¨G, T ∗α(t)k

∂∂,

where, by construction, T ∗α(t)k ∈ KαC . But G ∈ LC , k ∈ KC implies¨Tα(t)G, k

∂∂=¨T (t)G, k

∂∂=¨G, T ∗(t)k

∂∂.

Hence, T ∗(t)k = T ∗α(t)k ∈ KαC , k ∈ KαC that proves the statement.

Remark 4.13. As a result, (4.28) implies that for any k0 ∈ D(L∗) the Cauchyproblem in KC

∂

∂tkt = L∗kt

kt∣∣t=0

= k0

(4.35)

has a unique mild solution: kt = T ∗(t)k0 = T�(t)k0 ∈ D(L∗). Moreover,k0 ∈ KαC implies kt ∈ KαC provided (4.32) is satisfied.

49

Remark 4.14. The Cauchy problem (4.35) is well-posed in KC = D(L∗), i.e.,

for every k0 ∈ D(L�) there exists a unique solution kt ∈ KC of (4.35).

Let (4.28) and (4.32) be satisfied and let α0 be chosen as in the proof ofProposition 4.12 and fixed. Suppose that α ∈ (α0; 1). Then, Propositions 4.11

and 4.12 imply KαC ⊂ D(L∗) and the Banach subspace KαC is T ∗(t)- and,

therefore, T�(t)-invariant due to the continuity of these operators.

Let now T�α(t) be the restriction of the strongly continuous semigroup T�(t)

onto the closed linear subspace KαC . By general result (see, e.g., [19]), T�α(t)

is a strongly continuous semigroups on KαC with generator L�α which is therestriction of the operator L�. Namely,

D(L�α) ={k ∈ KαC

∣∣∣ L∗k ∈ KαC}, (4.36)

andL�αk = L�k = L∗k, k ∈ D(L�α) (4.37)

Since T (t) is a contraction semigroup on LC , then, T ′(t) is also a contraction

semigroup on (LC)′; but isomorphism (3.43) is isometrical, therefore, T ∗(t) is a

contraction semigroup on KC . As a result, its restriction T�α(t) is a contractionsemigroup on KαC . Note also, that by (4.36),

DαC :={k ∈ KαC

∣∣∣ L∗k ∈ KαC}is a core for L�α in KαC .

By (4.12), for any k ∈ KαC , G ∈ Bbs(Γ0) we have∫Γ0

(“PδG) (η) k (η) dλ (η)

=

∫Γ0

∑ξ⊂η

(1− δ)|ξ|∫

Γ0

(zδ)|ω|G (ξ ∪ ω)

∏y∈ξ

e−Eφ(y,ω)

×∏y∈η\ξ

Äe−E

φ(y,ω) − 1ädλ (ω) k (η) dλ (η)

=

∫Γ0

∫Γ0

(1− δ)|ξ|∫

Γ0

(zδ)|ω|G (ξ ∪ ω)

∏y∈ξ

e−Eφ(y,ω)

×∏y∈η

Äe−E

φ(y,ω) − 1ädλ (ω) k (η ∪ ξ) dλ (ξ) dλ (η)

=

∫Γ0

∫Γ0

∑ω⊂ξ

(1− δ)|ξ\ω| (zδ)|ω|G (ξ)∏y∈ξ\ω

e−Eφ(y,ω)

×∏y∈η

Äe−E

φ(y,ω) − 1äk (η ∪ ξ \ ω) dλ (ξ) dλ (η) ,

50

therefore,

(“P ∗δ k) (η) =∑ω⊂η

(1− δ)|η\ω| (zδ)|ω|∏

y∈η\ω

e−Eφ(y,ω) (4.38)

×∫

Γ0

∏y∈ξ

Äe−E

φ(y,ω) − 1äk (ξ ∪ η \ ω) dλ (ξ) .

Proposition 4.15. Suppose that (4.28) and (4.32) are fulfilled. Then, for anyk ∈ DαC and α ∈ (α0, 1), where α0 is chosen as in the proof of Proposition 4.12,

limδ→0

∥∥∥∥1

δ(“P ∗δ − 11)k − L�αk

∥∥∥∥KC

= 0. (4.39)

Proof. Let us recall (4.31) and define

(“P ∗,(0)δ k) (η) = (1− δ)(n)k(η);

(“P ∗,(1)δ k) (η) = zδ

∑x∈η

(1− δ)|η|−1eλÄe−φ(x−·), η \ x

ä×∫

Γ0

eλÄe−φ(x−·) − 1, ξ

äk (ξ ∪ η \ x) dλ (ξ) ;

and “P ∗,(≥2)δ = “P ∗δ − “P ∗,(0)

δ − “P ∗,(1)δ .

We will use the following elementary inequality, for any n ∈ N∪{0}, δ ∈ (0; 1)

0 ≤ n− 1− (1− δ)n

δ≤ δ n(n− 1)

2.

Then, for any k ∈ KαC and λ-a.a. η ∈ Γ0, η 6= ∅

C−|η|∣∣∣∣1δ (“P ∗,(0)

δ,ε − 11)k(η) + |η|k(η)

∣∣∣∣≤‖k‖KαCα|η|

∣∣∣∣|η| − 1− (1− δ)|η|

δ

∣∣∣∣ ≤ δ

2‖k‖KαCα|η||η|(|η| − 1) (4.40)

and the function αxx(x − 1) is bounded for x ≥ 1, α ∈ (0; 1). Next, for anyk ∈ KαC and λ-a.a. η ∈ Γ0, η 6= ∅

C−|η|∣∣∣∣1δ “P ∗,(1)

δ k(η)− z∑x∈η

∫Γ0

eλÄe−φ(x−·), η \ x

ä× eλ

Äe−φ(x−·) − 1, ξ

äk (ξ ∪ η \ x) dλ(ξ)

∣∣∣∣≤‖k‖KαC

z

αCα|η|

∑x∈η

(1− (1− δ)|η|−1) ∫

Γ0

eλÄαC(e−φ(x−·) − 1

), ξädλ (ξ)

≤‖k‖KαCz

αCα|η|

∑x∈η

(1− (1− δ)|η|−1)

exp {αCCφ}

≤‖k‖KαCz

αCα|η|δ|η|(|η| − 1) exp {αCCφ}. (4.41)

51

which is small in δ uniformly by |η|. Now, using inequality

1− e−Eφ(y,ω) = 1−

∏x∈ω

e−φ(x−y) ≤∑x∈ω

Ä1− e−φ(x−y)

ä,

we obtain

1

δC−|η|

∑ω⊂η|ω|≥2

(1− δ)|η\ω| (zδ)|ω| eλÄe−E

φ(·,ω), η \ ωä

×∫

Γ0

eλ

(∣∣∣e−Eφ(·,ω) − 1∣∣∣, ξ) |k(ξ ∪ η \ ω)|dλ (ξ)

= ‖k‖KαCα|η|1

δ

∑ω⊂η|ω|≥2

(1− δ)|η\ω|Åzδ

αCexp {αCCφ}

ã|ω|;

recall that α > α0, therefore, z exp{αCCφ} ≤ αC, and one may continue

≤‖k‖KαCα|η|1

δ

∑ω⊂η|ω|≥2

(1− δ)|η\ω| δ|ω|

= ‖k‖KαC δα|η||η|∑k=2

|η|!k! (|η| − k)!

(1− δ)|η|−k δk−2

= ‖k‖KαC δα|η||η|−2∑k=0

|η|!(k + 2)! (|η| − k − 2)!

(1− δ)|η|−k−2δk

= ‖k‖KαC δα|η| |η| (|η| − 1)

|η|−2∑k=0

(|η| − 2)!

(k + 2)! (|η| − k − 2)!(1− δ)|η|−2−k

δk

≤‖k‖KαC δα|η| |η| (|η| − 1)

|η|−2∑k=0

(|η| − 2)!

k! (|η| − k − 2)!(1− δ)|η|−2−k

δk

= ‖k‖KαC δα|η| |η| (|η| − 1) . (4.42)

Combining inequalities (4.40)–(4.42) we obtain (4.39).

As a result, we obtain an approximation for the semigroup.

Theorem 4.16. Let α0 be chosen as in the proof of the Proposition 4.12 andbe fixed. Let α ∈ (α0; 1) and k ∈ KαC be given. Then

(“P ∗δ )[t/δ]k → T�α(t)k, δ → 0

in the space KαC with norm ‖·‖KC for all t ≥ 0 uniformly on bounded intervals.

Proof. We may apply Proposition 4.15 to use Lemma 4.7 in the case Ln = L =LαC , pn = 11, fn = f = k, εn = δ → 0, n ∈ N.

52

4.5 Positive definiteness

We consider a small modification of the notion of positive definite functionsconsidered in Proposition 2.12. Namely, we denote by L0

ls(Γ0) the set of allmeasurable functions on Γ0 which have a local support, i.e. G ∈ L0

ls(Γ0) if thereexists Λ ∈ Bb(Rd) such that G �Γ0\Γ(Λ)= 0. We will say that a measurablefunction k : Γ0 → R is a positive defined function if, for any G ∈ L0

ls(Γ0) suchthat KG ≥ 0 and G ∈ LC for some C > 1 the inequality (2.30) holds.

For a given C > 1, we set LlsC = L0

ls(Γ0) ∩ LC . Since Bbs(Γ0) ⊂ LlsC , for any

C > 1, Proposition 2.12 (see also the second part of Remark 2.13) implies thatif k is a positive definite function as above then there exists a unique measureµ ∈M1

fm(Γ) such that k = kµ be its correlation function in the sense of (2.24).

Our aim is to show that the evolution k 7→ T�(t)k preserves this property ofthe positive definiteness.

Theorem 4.17. Let (4.28) holds and k ∈ D(L∗) ⊂ KC be a positive definite

function. Then kt := T�(t)k ∈ D(L∗) ⊂ KC will be a positive definite functionfor any t ≥ 0.

Proof. Let C > 0 be arbitrary and fixed. For any G ∈ LlsC we have∫

Γ0

G (η) kt (η) dλ (η) =

∫Γ0

(T (t)G) (η) k (η) dλ (η) . (4.43)

By Proposition 4.10, under condition (4.28), we obtain that

limn→0

∫Γ(Λn)

∣∣∣T [nt]n 11Γ(Λn)G (η)− 11Γ(Λn)(η)(T (t)G) (η)

∣∣∣C |η|dλ (η) = 0,

where for n ≥ 2Tn = “PΛn

1n

and Λn ↗ Rd. Note that, by the dominated convergence theorem,∫Γ0

(T (t)G) (η) k (η) dλ (η) = limn→∞

∫Γ0

11Γ(Λn) (η) (T (t)G) (η) k (η) dλ (η)

= limn→∞

∫Γ(Λn)

(T (t)G) (η) k (η) dλ (η) .

Next,∣∣∣∣∣∫

Γ(Λn)

(T (t)G) (η) k (η) dλ (η)−∫

Γ(Λn)

T [nt]n 11Γ(Λn)G (η) k (η) dλ (η)

∣∣∣∣∣≤∫

Γ(Λn)

∣∣∣T [nt]n 11Γ(Λn)G (η)− 11Γ(Λn)(η)(T (t)G) (η)

∣∣∣ k (η) dλ (η)

≤‖k‖KC∫

Γ(Λn)

∣∣∣T [nt]n 11Γ(Λn)G (η)− 11Γ(Λn)(η)(T (t)G) (η)

∣∣∣C |η|dλ (η)→ 0, n→∞.

53

Therefore,∫Γ0

(T (t)G) (η) k (η) dλ (η) = limn→∞

∫Γ(Λn)

T [nt]n 11Γ(Λn)G (η) k (η) dλ (η) . (4.44)

Our aim is to show that for any G ∈ LlsC the inequality KG ≥ 0 implies∫

Γ0

G (η) kt (η) dλ (η) ≥ 0.

By (4.43) and (4.44), it is enough to show that for any m ∈ N and for anyG ∈ Lls

C such that KG ≥ 0 the following inequality holds∫Γ0

11Γ(Λn)Tmn 11Γ(Λn)G (η) k (η) dλ (η) ≥ 0, m ∈ N0. (4.45)

The inequality (4.45) is fulfilled if only

K11Γ(Λn)Tmn Gn ≥ 0, (4.46)

where Gn := 11Γ(Λn)G. Note that(K11Γ(Λn)T

mn Gn

)(γ) =

∑ηbγ

11Γ(Λn) (η) (Tmn Gn) (η) (4.47)

=∑η⊂γΛn

(Tmn Gn) (η) = (KTmn Gn) (γΛn)

for any m ∈ N0. In particular,

(KGn) (γ) =(K11Γ(Λn)G

)(γ) = (KG) (γΛn) ≥ 0. (4.48)

Let us now consider any G ∈ LlsC (stress that G is not necessary equal to 0

outside of Γ(Λn)) and suppose that(KG

)(γ) ≥ 0 for any γ ∈ Γ(Λn). Then(

KTnG)

(γΛn) =(K“PΛn

1n

G)

(γΛn) =(PΛn

1n

KG)

(γΛn) (4.49)

=(

ΞΛn1n

(γΛn))−1 ∑

η⊂γΛn

Å1

n

ã|η|Å1− 1

n

ã|γ\η|×∫

Γ(Λn)

Åz

n

ã|ω|∏y∈ω

e−Eφ(y,γ)

(KG

)((γΛn \ η) ∪ ω

)dλ (ω) ≥ 0.

By (4.48), setting G = Gn ∈ LlsC we obtain, because of (4.49), KTnGn ≥ 0.

Next, setting G = TnGn ∈ LlsC we obtain, by (4.49), KT 2

nGn ≥ 0. Then, usingan induction mechanism, we obtain that

(KTmn Gn) (γΛn) ≥ 0, m ∈ N0,

that, by (4.46) and (4.47), yields (4.45). This completes the proof.

54

4.6 Ergodicity

Let k ∈ KαC be such that k(∅) = 0 then, by (4.38), (“P ∗δ k) (∅) = 0. Class of allsuch functions we denote by K0

α.

Proposition 4.18. Assume that there exists ν ∈ (0; 1) such that

z ≤ min{νCe−CCφ ; 2Ce−2CCφ

}. (4.50)

Let, additionally, α ∈ (α0; 1), where α0 is chosen as in the proof of the Propo-sition 4.12. Then for any δ ∈ (0; 1) the following estimate holds∥∥∥“P ∗δ �K0

α

∥∥∥ ≤ 1− (1− ν)δ. (4.51)

Proof. It is easily seen that for any k ∈ K0α the following inequality holds

|k (η)| ≤ 1|η|>0 ‖k‖KC C|η|, λ−a.a. η ∈ Γ0.

Then, using (4.38), we have

C−|η|∣∣∣(“P ∗δ k) (η)

∣∣∣≤C−|η|

∑ω⊂η

(1− δ)|η\ω| (zδ)|ω|∫

Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)ä|k (ξ ∪ η \ ω)| dλ (ξ)

≤‖k‖KC∑ω⊂η

(1− δ)|η\ω|Åzδ

C

ã|ω| ∫Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)äC |ξ|11|ξ|+|η\ω|>0dλ (ξ)

= ‖k‖KC∑ω(η

(1− δ)|η\ω|Åzδ

C

ã|ω| ∫Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)äC |ξ|dλ (ξ)

+ ‖k‖KC

Åzδ

C

ã|η| ∫Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)äC |ξ|11|ξ|>0dλ (ξ)

= ‖k‖KC∑ω(η

(1− δ)|η\ω|Åzδ

C

ã|ω| ∫Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)äC |ξ|dλ (ξ)

+ ‖k‖KC

Åzδ

C

ã|η| ∫Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)äC |ξ|dλ (ξ)− ‖k‖KC

Åzδ

C

ã|η|= ‖k‖KC

∑ω⊂η

(1− δ)|η\ω|Åzδ

C

ã|ω| ∫Γ0

∏y∈ξ

Ä1− e−E

φ(y,ω)äC |ξ|dλ (ξ)

− ‖k‖KC

Åzδ

C

ã|η|= ‖k‖KC

∑ω⊂η

(1− δ)|η\ω|Åzδ

C

ã|ω|exp

ßC

∫Rd

Ä1− e−E

φ(y,ω)ädy

™55

− ‖k‖KC

Åzδ

C

ã|η|≤‖k‖KC

∑ω⊂η

(1− δ)|η\ω|Åzδ

C

ã|ω|exp {CCβ |ω|} − ‖k‖KC

Åzδ

C

ã|η|≤‖k‖KC

∑ω⊂η

(1− δ)|η\ω| (νδ)|ω| − ‖k‖KC

Åzδ

C

ã|η|= ‖k‖KC

Ç(1− (1− ν) δ)

|η| −Åzδ

C

ã|η|å= ‖k‖KC

Å1− (1− ν) δ − zδ

C

ã |η|−1∑j=0

(1− (1− ν) δ)|η|−1−|j|

Åzδ

C

ãj≤‖k‖KC

Å1− (1− ν) δ − zδ

C

ã |η|−1∑j=0

Åzδ

C

ãj= ‖k‖KC

Å1− (1− ν) δ − zδ

C

ã1−

(zδC

)|η|1− zδ

C

≤‖k‖KC

Å1− (1− ν) δ − zδ

C

ã1

1− zδC

= ‖k‖KC

Ç1− (1− ν) δ

1− zδC

å≤ ‖k‖KC

(1− (1− ν) δ

),

where we have used that, clearly, z < νC < C. The statement is proved.

Remark 4.19. Condition (4.50) is equivalent to (4.28) and (4.32).

As it was mentioned in Example 3.18, under condition (cf. (4.33))

zCφ < (2e)−1, (4.52)

there exists (see, e.g., [35] for details) a Gibbs measure µ on(Γ,B(Γ)

)cor-

responding to the potential φ ≥ 0 and activity parameter z. We denote thecorresponding correlation function by kµ. The measure µ is reversible (sym-metrizing) for the operator defined by (4.1) (see, e.g., [35, 54]). Therefore, forany F ∈ KBbs(Γ0) ∫

Γ

LF (γ)dµ(γ) = 0. (4.53)

Theorem 4.20. Let (4.52) and (4.50) hold and let α ∈ (α0; 1), where α0 is

chosen as in the proof of Proposition 4.12. Let k0 ∈ KαC , kt = T�α(t)k0. Thenfor any t ≥ 0

‖kt − kµ‖KC ≤ e−(1−ν)t‖k0 − kµ‖KC . (4.54)

Proof. First of all, let us note that for any α ∈ (α0; 1) the inequality (4.34)implies z ≤ αC exp{−αCCφ}. Hence kµ(η) ≤ (αC)|η|, η ∈ Γ0. Therefore, kµ ∈

56

KαC ⊂ KαC ∩D(L∗). By (4.53), for any G ∈ Bbs(Γ0) we have 〈〈LG, kµ〉〉 = 0.

It means that L∗kµ = 0. Therefore, L�αkµ = 0. As a result, T�α(t)kµ = kµ.Let r0 = k0 − kµ ∈ KαC . Then r0 ∈ K0

a and

‖kt − kµ‖KC =∥∥T�α(t)r0

∥∥KC

≤∥∥∥(“P ∗δ )[ tδ ]r0

∥∥∥KC

+∥∥∥T�α(t)r0 −

(“P ∗δ )[ tδ ]r0

∥∥∥KC

≤∥∥∥“P ∗δ �K0

α

∥∥∥[ tδ ] · ‖r0‖KC +∥∥∥T�α(t)r0 −

(“P ∗δ )[ tδ ]r0

∥∥∥KC

≤(1− (1− ν)δ

) tδ−1‖r0‖KC +

∥∥∥T�α(t)r0 −(“P ∗δ )[ tδ ]r0

∥∥∥KC,

since 0 < 1 − (1 − ν)δ < 1 and tδ <

[tδ

]+ 1. Taking the limit as δ ↓ 0 in the

right hand side of this inequality we obtain (4.54).

References

[1] S. Albeverio, Y. Kondratiev, and M. Rockner. Analysis and geometry onconfiguration spaces. J. Funct. Anal., 154(2):444–500, 1998.

[2] S. Albeverio, Y. Kondratiev, and M. Rockner. Analysis and geometry onconfiguration spaces: the Gibbsian case. J. Funct. Anal., 157(1):242–291,1998.

[3] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas,R. Nagel, F. Neubrander, and U. Schlotterbeck. One-parameter semi-groups of positive operators, volume 1184 of Lecture Notes in Mathematics.Springer-Verlag, Berlin, 1986. ISBN 3-540-16454-5. x+460 pp.

[4] Y. Berezansky, Y. Kondratiev, T. Kuna, and E. Lytvynov. On a spec-tral representation for correlation measures in configuration space analysis.Methods Funct. Anal. Topology, 5(4):87–100, 1999.

[5] C. Berns, Y. Kondratiev, Y. Kozitsky, and O. Kutoviy. Kawasaki dynamicsin continuum: micro- and mesoscopic descriptions. J. of Dyn and Diff.Eqn., 25(4):1027–1056, 2013.

[6] L. Bertini, N. Cancrini, and F. Cesi. The spectral gap for a Glauber-typedynamics in a continuous gas. Ann. Inst. H. Poincare Probab. Statist., 38(1):91–108, 2002.

[7] N. N. Bogoliubov. Problems of a dynamical theory in statistical physics.In Studies in Statistical Mechanics, Vol. I, pages 1–118. North-Holland,Amsterdam, 1962.

[8] B. Bolker and S. W. Pacala. Using moment equations to understandstochastically driven spatial pattern formation in ecological systems. Theor.Popul. Biol., 52(3):179–197, 1997.

57

[9] B. Bolker and S. W. Pacala. Spatial moment equations for plant com-petitions: Understanding spatial strategies and the advantages of shortdispersal. American Naturalist, 153:575–602, 1999.

[10] A. Borodin and G. Olshanski. Point processes and the infinite symmetricgroup. Math. Res. Lett., 5(6):799–816, 1998.

[11] A. Borodin and G. Olshanski. Distributions on partitions, point processes,and the hypergeometric kernel. Comm. Math. Phys., 211(2):335–358, 2000.

[12] A. Borodin and G. Olshanski. Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes. Ann. ofMath. (2), 161(3):1319–1422, 2005.

[13] A. Borodin and G. Olshanski. Representation theory and random pointprocesses. In European Congress of Mathematics, pages 73–94. Eur. Math.Soc., Zurich, 2005.

[14] A. Borodin, G. Olshanski, and E. Strahov. Giambelli compatible pointprocesses. Adv. in Appl. Math., 37(2):209–248, 2006.

[15] N. R. Campbell. The study of discontinuous problem. Proc. CambridgePhilos. Soc., 15:117–136, 1909.

[16] N. R. Campbell. Discontinuities in light emission. Proc. Cambridge Philos.Soc., 15:310–328, 1910.

[17] U. Dieckmann and R. Law. Relaxation projections and the method of mo-ments. In The Geometry of Ecological Interactions, pages 412–455. Cam-bridge University Press, Cambridge, UK, 2000.

[18] R. L. Dobrushin, Y. G. Sinai, and Y. M. Sukhov. Dynamical systemsof statistical mechanics. In Y. G. Sinai, editor, Ergodic Theory with Ap-plications to Dynamical Systems and Statistical Mechanics, volume II ofEncyclopaedia Math. Sci., Berlin, Heidelberg, 1989. Springer.

[19] K.-J. Engel and R. Nagel. One-parameter semigroups for linear evolutionequations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag,New York, 2000. ISBN 0-387-98463-1. xxii+586 pp.

[20] S. N. Ethier and T. G. Kurtz. Markov processes: Characterization and con-vergence. Wiley Series in Probability and Mathematical Statistics: Prob-ability and Mathematical Statistics. John Wiley & Sons Inc., New York,1986. ISBN 0-471-08186-8. x+534 pp.

[21] D. Filonenko, D. Finkelshtein, and Y. Kondratiev. On two-componentcontact model in continuum with one independent component. MethodsFunct. Anal. Topology, 14(3):209–228, 2008.

[22] D. Finkelshtein. Measures on two-component configuration spaces. Con-densed Matter Physics, 12(1):5–18, 2009.

58

[23] D. Finkelshtein. Functional evolutions for homogeneous stationary death-immigration spatial dynamics. Methods Funct. Anal. Topology, 17(4):300–318, 2011.

[24] D. Finkelshtein and Y. Kondratiev. Measures on configuration spaces de-fined by relative energies. Methods Funct. Anal. Topology, 11(2):126–155,2005.

[25] D. Finkelshtein, Y. Kondratiev, and Y. Kozitsky. Glauber dynamics incontinuum: a constructive approach to evolution of states. Discrete andCont. Dynam. Syst. - Ser A., 33(4):1431–1450, 4 2013.

[26] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Individual based modelwith competition in spatial ecology. SIAM J. Math. Anal., 41(1):297–317,2009.

[27] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for theGlauber dynamics in continuum. Infin. Dimens. Anal. Quantum Probab.Relat. Top., 14(4):537–569, 2011.

[28] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Correlation functionsevolution for the Glauber dynamics in continuum. Semigroup Forum, 85:289–306, 2012.

[29] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Semigroup approach tobirth-and-death stochastic dynamics in continuum. J. of Funct. Anal., 262(3):1274–1308, 2012.

[30] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Establishment and fe-cundity in spatial ecological models: statistical approach and kinetic equa-tions. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 16(2):1350014(24 pages), 2013.

[31] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. An operator approach toVlasov scaling for some models of spatial ecology. Methods Funct. Anal.Topology, 19(2):108–126, 2013.

[32] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Binaryjumps in continuum. I. Equilibrium processes and their scaling limits.J. Math. Phys., 52:063304:1–25, 2011.

[33] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Binaryjumps in continuum. II. Non-equilibrium process and a Vlasov-type scalinglimit. J. Math. Phys., 52:113301:1–27, 2011.

[34] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Zhizhina. An approx-imative approach for construction of the Glauber dynamics in continuum.Math. Nachr., 285(2–3):223–235, 2012.

59

[35] D. Finkelshtein, Y. Kondratiev, and E. Lytvynov. Equilibrium Glauberdynamics of continuous particle systems as a scaling limit of Kawasakidynamics. Random Oper. Stoch. Equ., 15(2):105–126, 2007.

[36] D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Markov evolutionsand hierarchical equations in the continuum. I. One-component systems.J. Evol. Equ., 9(2):197–233, 2009.

[37] D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Glauber dynamics inthe continuum via generating functionals evolution. Complex Analysis andOperator Theory, 6(4):923–945, 2012.

[38] D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Kawasaki dynamics inthe continuum via generating functionals evolution. Methods Funct. Anal.Topology, 18(1):55–67, 2012.

[39] D. Finkelshtein, Y. Kondratiev, and M. J. Oliveira. Markov evolutionsand hierarchical equations in the continuum. II: Multicomponent systems.Reports Math. Phys., 71(1):123–148, 2013.

[40] M. E. Fisher and D. Ruelle. The stability of many-particle systems.J. Math. Phys., 7:260–270, 1966.

[41] N. Fournier and S. Meleard. A microscopic probabilistic description of alocally regulated population and macroscopic approximations. The Annalsof Applied Probability, 14(4):1880–1919, 2004.

[42] N. L. Garcia. Birth and death processes as projections of higher dimensionalpoisson processes. Adv. in Appl. Probab., 27:911–930., 1995.

[43] N. L. Garcia and T. G. Kurtz. Spatial birth and death processes as solutionsof stochastic equations. ALEA Lat. Am. J. Probab. Math. Stat., 1:281–303(electronic), 2006.

[44] N. L. Garcia and T. G. Kurtz. Spatial point processes and the projectionmethod. Progress in Probability, 60:271–298, 2008.

[45] H.-O. Georgii. Canonical and grand canonical Gibbs states for continuumsystems. Comm. Math. Phys., 48(1):31–51, 1976.

[46] R. A. Holley and D. W. Stroock. Nearest neighbor birth and death processeson the real line. Acta Math., 140(1-2):103–154, 1978.

[47] D. G. Kendall. Stochastic Geometry, chapter Foundations of a theory ofrandom sets, pages 322–376. New York: Wiley,, 1974.

[48] Y. Kondratiev and T. Kuna. Harmonic analysis on configuration space. I.General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5(2):201–233, 2002.

60

[49] Y. Kondratiev and T. Kuna. Correlation functionals for Gibbs measuresand Ruelle bounds. Methods Funct. Anal. Topology, 9(1):9–58, 2003.

[50] Y. Kondratiev and O. Kutoviy. On the metrical properties of the configu-ration space. Math. Nachr., 279(7):774–783, 2006.

[51] Y. Kondratiev, O. Kutoviy, and R. Minlos. On non-equilibrium stochasticdynamics for interacting particle systems in continuum. J. Funct. Anal.,255(1):200–227, 2008.

[52] Y. Kondratiev, O. Kutoviy, and S. Pirogov. Correlation functions and in-variant measures in continuous contact model. Infin. Dimens. Anal. Quan-tum Probab. Relat. Top., 11(2):231–258, 2008.

[53] Y. Kondratiev, O. Kutoviy, and E. Zhizhina. Nonequilibrium Glauber-typedynamics in continuum. J. Math. Phys., 47(11):113501, 17, 2006.

[54] Y. Kondratiev and E. Lytvynov. Glauber dynamics of continuous particlesystems. Ann. Inst. H. Poincare Probab. Statist., 41(4):685–702, 2005.

[55] Y. Kondratiev, E. Lytvynov, and M. Rockner. Equilibrium Kawasaki dy-namics of continuous particle systems. Infin. Dimens. Anal. QuantumProbab. Relat. Top., 10(2):185–209, 2007.

[56] Y. Kondratiev, R. Minlos, and E. Zhizhina. One-particle subspace of theGlauber dynamics generator for continuous particle systems. Rev. Math.Phys., 16(9):1073–1114, 2004.

[57] T. Kuna. Studies in configuration space analysis and applications. BonnerMathematische Schriften [Bonn Mathematical Publications], 324. Univer-sitat Bonn Mathematisches Institut, Bonn, 1999. ii+187 pp. Dissertation,Rheinische Friedrich-Wilhelms-Universitat Bonn, Bonn, 1999.

[58] A. Lenard. Correlation functions and the uniqueness of the state in classicalstatistical mechanics. Comm. Math. Phys., 30:35–44, 1973.

[59] A. Lenard. States of classical statistical mechanical systems of infinitelymany particles. I. Arch. Rational Mech. Anal., 59(3):219–239, 1975.

[60] A. Lenard. States of classical statistical mechanical systems of infinitelymany particles. II. Characterization of correlation measures. Arch. RationalMech. Anal., 59(3):241–256, 1975.

[61] S. A. Levin. Complex adaptive systems: exploring the known, the unknownand the unknowable. Bulletin of the AMS, 40(1):3–19, 2002.

[62] T. M. Liggett. Interacting particle systems, volume 276 of Grundlehren derMathematischen Wissenschaften [Fundamental Principles of MathematicalSciences]. Springer-Verlag, New York, 1985. ISBN 0-387-96069-4. xv+488pp.

61

[63] T. M. Liggett. Stochastic interacting systems: contact, voter and ex-clusion processes, volume 324 of Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1999. ISBN 3-540-65995-1. xii+332 pp.

[64] H. P. Lotz. Uniform convergence of operators on L∞ and similar spaces.Math. Z., 190(2):207–220, 1985.

[65] J. Mecke. Eine charakteristische Eigenschaft der doppelt stochastischenPoissonschen Prozesse. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete,11:74–81, 1968.

[66] U. Murrell, David J. and Dieckmann and R. Law. On moment closures forpopulation dynamics in continuous space. Journal of Theoretical Biology,229(3):421 – 432, 2004.

[67] X.-X. Nguyen and H. Zessin. Integral and differential characterizations ofthe Gibbs process. Math. Nachr., 88:105–115, 1979.

[68] O. Ovaskainen, D. Finkelshtein, O. Kutoviy, S. Cornell, B. Bolker, andY. Kondratiev. A mathematical framework for the analysis of spatial-temporal point processes. Theoretical Ecology, 2013.

[69] L. V. Ovsjannikov. Singular operator in the scale of Banach spaces. Dokl.Akad. Nauk SSSR, 163:819–822, 1965.

[70] A. Pazy. Semigroups of linear operators and applications to partial differ-ential equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. ISBN 0-387-90845-5. viii+279 pp.

[71] M. D. Penrose. Existence and spatial limit theorems for lattice and contin-uum particle systems. Prob. Surveys, 5:1–36, 2008.

[72] R. S. Phillips. The adjoint semi-group. Pacific J. Math., 5:269–283, 1955.

[73] C. Preston. Spatial birth-and-death processes. Bull. Inst. Internat. Statist.,46(2):371–391, 405–408, 1975.

[74] A. Renyi. Remarks on the Poisson process. Studia Sci. Math. Hungar., 2:119–123, 1970.

[75] D. Ruelle. Statistical mechanics: Rigorous results. W. A. Benjamin, Inc.,New York-Amsterdam, 1969. xi+219 pp.

[76] D. Ruelle. Superstable interactions in classical statistical mechanics.Comm. Math. Phys., 18:127–159, 1970.

[77] M. V. Safonov. The abstract Cauchy-Kovalevskaya theorem in a weightedBanach space. Comm. Pure Appl. Math., 48(6):629–637, 1995.

62

[78] F. Treves. Ovcyannikov theorem and hyperdifferential operators. Notas deMatematica, No. 46. Instituto de Matematica Pura e Aplicada, ConselhoNacional de Pesquisas, Rio de Janeiro, 1968. iii+238 pp.

[79] J. van Neerven. The adjoint of a semigroup of linear operators, volume 1529of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1992. ISBN 3-540-56260-5. x+195 pp.

63